Lecture III of VI (Claudio Piani) Rotation, vorticity, geostrophic adjustment, inertial oscillations, gravity waves with rotation.

Rotation What happens when we are no longer on an inertial frame of reference? If we are on a rotating frame of reference we can use complex numbers to describe the change of reference systems. By taking the second derivative of the transformed coordinates we derive the Coriolis and centrifugal apparent forces (you should be able to recognize them).

Rotation In standard (x, y) notation the Coriolis term is given by: Given that the earth is more or less spherical, the  in the equation above is, in fact, the projection of the earths rotation (  ) onto the tangential plane at point P. Usually we use the term f = 2  in the equations

The simple case: shallow rotating pool Let us take the same shallow pool we had in the last lecture and add the Coriolis term to the RHS. h u v

shallow water: equations of motion All that fuss for two small extra terms?!?!?! But wait, see what they do! We can start by linearizing the SWE again about a state of no motion...

SWE: linearization with f Remember the terms in lowercase are the perturbations about the mean state. If you look at these equations you will see that now you can have a steady state solution with a bump! Think about it, this is not something your common sense will tell you… If you perturb the surface of the liquid it will not relax back to the original flat state! Instead a current will arise so that the Coriolis terms in the equation balance the height (pressure) term. This problem is called geostrophic adjustment and the final state is said to be in geostrophic balance. Before we enjoy solving this problem analytically we should introduce some important quantities that these equations preserve.

SWE: vorticity We define circulation of a fluid below (you should know what the symbols mean): And from Stoke’s theorem (which you vaguely remember): And of course the thing in parenthesis is the “curl” of the velocity. Since we are still only dealing with a shallow pool and horizontal motions, the curl of velocity is: This is generally referred to as “relative” vorticity, as in relative to the flow. Later we will see that there are other kinds of vorticity apart from the curl of the horizontal velocity.

SWE: vorticity To obtain a vorticity conservation equation we start by taking the curl of the momentum SWE and use the continuity equation to eliminate the divergence term… The final term in parenthesis is called potential vorticity and is conserved in the SWE. Notice that in the absence of rotation f is zero and SWE conserve relative vorticity alone. Well that was fun…. Now let’s get back to the geostrophic adjustment problem…

SWE: geostrophic adjustment If potential vorticity is conserved then we can calculated the final shape of the free surface h. Let us assume that h(t=0) is a step (black solid line in diagram, think of a dam breaking). After infinite time and at steady state we will have a balance between the height term (red dashed line) and the Coriolis associated with a meridional current (blue dashed line).

SWE: Geostrophic adjustment This is a second order ODE. You should know how to solve it, boundary conditions and all… h v

SWE: Geostrophic adjustment

Inertial oscillations Solving the geostrophic adjustment problem showed us that, at steady state, we can have an isolated bump in the surface of the shallow water. We will now see that, in the case of a rotating fluid, there is an entire class of fluid motions, called inertial oscillations, where the free surface of the fluid is fixed. Let’s go back to the RLSWE and assume steady state in the third equation: This is called a non divergent flow…

Inertial oscillations We are looking for the time dependent component of the solutions to the RLSWE where h does not vary in time this is also referred to the rigid lid approximation. Since h does not have a time varying component we will look for solutions to: We can simplify things by using complex number formalism and introduce a complex velocity: The solution is simply:

Inertial oscillations What do these motions look like? The web, a.k.a. wikipedia, is full of examples, conceptual, experimental, numerical, observational, etc… of inertial oscillations, go nuts… look at them all….

LSWE: gravity waves with f As we did before, we can put in a solution in the form of a wave and derive the following dispersion relation :

Exercise: 1.Starting from the RSWE in the last slide derive the dispersion relation for gravity waves (also given in the last slide). 2.Qualitatively describe the geostrophic adjustment process of an initial meridional current confined to a band centered on the x=0 axis. 3.Derive the 4 components of the ray tracing equations using the dispersion relation given on the last slide.