Second Midterm

• 1
  Full Dimensional Analysis: Introducing Rossby and Froude Numbers

  Consider the rotating, stratified Boussinesq equations:

  $$\begin{aligned} \frac{D\boldsymbol{u}}{Dt} + 2\boldsymbol{\Omega... ...ac{Db}{Dt} + N^2 w &= 0, \\ \nabla \cdot \boldsymbol{u} &= 0. \end{aligned}$$
  Here $b$ is buoyancy and $N$ is the Brunt–Väisälä frequency.

  (a)

  Introduce characteristic scales $U, L, W, H$ and time scale $T$ to write the non-dimensional form of the equations.

  (b)

  Identify and define all resulting non-dimensional parameters:

  $$Ro = \frac{U}{fL}, \qquad Fr = \frac{U}{N H}, \qquad \delta = \frac{H}{L}.$$

  Briefly explain the physical meaning of each:

  • $Ro$ (Rossby number) measures the ratio of inertial to Coriolis forces,
  • $Fr$ (Froude number) measures the ratio of inertial to buoyancy (stratification) forces,
  • $\delta$ (aspect ratio) measures the thinness of the flow domain.

  (c)

  Assuming $Ro\ll1$ and $Fr \ll 1$, simplify the dimensionless momentum equations and state the leading-order balances that emerge.

  (d)

  Estimate the Rossby number for each of the following flows:

  (i)

  A midlatitude ocean current with $U = 0.1\,\mathrm{m/s}$ and $L = 100\,\mathrm{km}$.

  (ii)

  A tornado with $U = 50\,\mathrm{m/s}$ and $L = 100\,\mathrm{m}$.

  Take $f = 10^{-4}\,\mathrm{s}^{-1}$. Which flow is more strongly influenced by rotation, and why?

  (e)

  Conceptual: Explain briefly why small Rossby number implies that large-scale flows tend to be nearly two-dimensional along the rotation axis. Relate this to your answer in part (c).

SOLUTIONS