The vertical and horizontal structure of steady forced Rossby waves depends on the velocity and stratification profiles in the basic state. This demonstration allows you to explore this dependence.
1: The model
We consider the linearised quasi-geostrophic equations in a -plane. and are horizontal coordinates representing longitude and latitude respectively and represents height. The channel is assumed to have rigid walls at , where the boundary condition of no normal flow is applied. The basic state is one of flow with speed in the -direction and the backgound stratification is represented by the buoyancy frequency . The flow is disturbed by topographic forcing at the lower boundary, with topographic height of the form
The disturbances to the basic flow are represented
by a quasi-geostrophic streamfunction , assumed to have
The equation for
may be derived from the linearised form
quasi-geostrophic potential vorticity equation, and
takes the form
2: Running the demonstration
On entering this demonstration you must:
(i) specify either a topographic shape and basic state distributions of and , by selecting Specify formulae.
(ii) select Calculate to determine the steady state wave field. This is depicted in terms of the distribution of , i.e. of the streamfunction along the centre of the channel, in the plane.
(iii) Other menu options control some of the internal parameters of the calculation and are described in detail in §4 below.
The method of solution is to take the Fourier transform of (17.1) with respect to . A set of ordinary differential equations in for each Fourier component results, and each may be solved using a simple numerical scheme. An inverse transform may then be applied to the resulting expressions to give the solution for as a function of and .
The method of solution implicitly assumes that the -domain is periodic. You must therefore remember that when you specify an initial distribution, or a forcing distribution, that distribution is actually replicated periodically an infinite number of times to the left and to the right of your field of view. If you want to use this demonstration to study the response to an isolated topographic forcing, you must therefore choose the damping constant to be sufficiently large that the waves have decayed to a small factor of their original amplitude in the time it takes the waves to propagate through the length of the domain.
Otherwise, the method of solution is extremely robust.
4: Menu Options
Note that most of the parameters for this demonstration are not displayed on the screen initially. To inspect the model parameters, functions or grid size, choose the appropriate menu option and the information will be displayed in a window. Hitting escape in these windows leaves the data unchanged and returns to the menu.
Note that this demonstration works in SI units. Sensible default values are given for the various parameters.
Calculate: Evaluates and draws a contour plot of psi.
Model: Allows you to specify the parameters , , and .
Specify formulae: allows you to change the formulae used for , and .
Domain: allows you to specify the ranges of and values you are interested in.
Accuracy: allows you to set the number of grid points used in the and directions. The number of points in the direction must be a power of 2. The maximum number of points in either direction is 64.
Exit: returns to the GEFD main menu.
5: Suggested experiments
(i) Taking sinusoidal topography and independent of height, you might verify the basic predictions of the Charney-Drazin model, viz. vertical propagation when the basic velocity is positive and not too strong, vertical trapping otherwise. Does the critical positive velocity, above which there is no vertical propagation, have the expected dependence on horizontal wavelength?
(ii) Keeping sinusoidal topography, now let increase with height, starting below the critical velocity and increasing to above it. What is the vertical structure of the waves? Now change the topography to an isolated hump. Can you explain the pattern that results? You should find that some waves excited by the topography are trapped in the vertical, whilst others propagate.
(iii) Now try varying the stratification, e.g. let increase with height as it does from troposphere to stratosphere, in the real atmosphere. Could the increase in from troposphere to stratosphere alone account for the observed structure of the waves with height?
Charney, J.G. and Eliassen, A., 1949: A numerical model for prediciting the perturbations of the middle latitude westerlies. Tellus, 1, 38-54.
White, A.A., 1982: Zonal translation properties of two quasi-geostrophic systems of equations. J. Atmos. Sci., 39, 2107-2118.