32 The Coriolis force and Rossby Number

In the inertial frame of reference (upper part of the picture), the black ball moves in a straight line. However, the observer (red dot) who is standing in the rotating/non-inertial frame of reference (lower part of the picture) sees the object as following a curved path due to the Coriolis and centrifugal forces present in this frame.

As you learned in introductory mechanics, particles in uniform circular motion are in a state of constant acceleration, because the direction of the velocity vector changes as a particle moves around the circle. Inside the rotating reference frame, there is an apparent force, called the centrifugal force, that it is directed outward with a magnitude equal to \Omega^2 r. Each particle of the Earth is also subject to a centrifugal force directed outward from the axis of rotation. However, that force is balanced by gravity in a slightly ellipsoidal body. Thus the planetary centrifugal force becomes part of the background state and does not enter our momentum equations.

However, if a particle is moving relative to the rotating reference frame, then another apparent force, called the Coriolis force, arises. Its magnitude is a product of the vorticity and the velocity relative to the reference frame. Its direction is perpendicular to the velocity vector. For ocean flows, the components of the Coriolis force due to the horizontal velocity components are by far the largest.

Key Takeaways

In oceanographic applications, to close approximation, the Coriolis force is

    \[ \underline{\textbf{F}}_{Coriolis} =\left( fv,-fu,0 \right) \]

The Coriolis force acts at a 90 degree angle to the velocity vector. Because f changes sign, in the Northern Hemisphere it is directed 90° to the right of the direction the velocity vector points, while in the Southern Hemisphere it is directed 90° to the left. When we include the Coriolis force, the horizontal components of the momentum equation become

    \[ \frac{Du}{Dt}=fv+\nu \nabla^2 u+\sum F^x \]

    \[ \frac{Dv}{Dt}=-fu+\nu \nabla^2 v+\sum F^y \]

Let’s expand the left side of the x-momentum equation (the Lagrangian rate of change) and further assume that since the vertical velocity is typically much smaller than the horizontal velocity, the term involving the vertical advection of momentum w \partial u / \partial z, is negligible:

    \[ \frac{\partial u}{\partial t} +u\frac{\partial u}{\partial x}+ v\frac{\partial u}{\partial y} =fv+\nu \nabla^2 u+\sum F^x. \]

The Rossby Number is a non-dimensional number expressing the ratio of the inertial (i.e., momentum advection) and Coriolis terms, given knowledge of order of magnitude sizes for the current speed (U), horizontal length scale (L) and Coriolis parameter (f, a simple function of the latitude). Both inertial terms involve the product of velocity and velocity gradient and either is expected to be size

    \[ u\frac{\partial u}{\partial x} \approx U\frac{U}{L}.  \]

The Coriolis force is the product of Coriolis parameter and the velocity and is expected to be size fU.

Key Takeaways

In scaling arguments, the ratio of the size of the inertial term(s) to the Coriolis term is called the Rossby Number

    \[Ro = \frac{U}{fL}  \]

If we wanted to compare the Eulerian acceleration term (\partial u / \partial t \approx U/T) to the Coriolis term based on a known time scale (T), we could define an analogous temporal Rossby Number

    \[Ro_t = \frac{1}{fT}.  \]

Usually time and space scales are related via U = L/T and so it is expected that these numbers are similar in a given physical situation.  If the Rossby number(s) are small then both acceleration terms can be neglected in a given problem.

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