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 $\beta$-plane. $x$ and $y$ are horizontal coordinates representing longitude and latitude respectively and $z$ represents height. The channel is assumed to have rigid walls at $y = \pm {\textstyle {1 \over 2}} L$, where the boundary condition of no normal flow is applied. The basic state is one of flow with speed $U (z)$ in the $x$-direction and the backgound stratification is represented by the buoyancy frequency $N(z)$. The flow is disturbed by topographic forcing at the lower boundary, with topographic height $h$ of the form $h ( x , y ) = \hat h ( x ) \cos ( \pi y / L )$

The disturbances to the basic flow are represented by a quasi-geostrophic streamfunction $\psi$, assumed to have the form $\psi ( x , y , z ) = \hat \psi ( x , z )\cos ( \pi y / L )$. The equation for $\hat \phi ( x , z )$ may be derived from the linearised form of the quasi-geostrophic potential vorticity equation, and takes the form

$$( \alpha + {\partial \over \partial x} ) \left ( \hat \phi_{... ...f^2 \over N^2} U_z )_z \right ) \hat \psi_x = 0 . \eqno (17.1)$$

where $\kappa_{h} = \pi / L$. The constant $\alpha$ represents the effect of constant damping on both temperature or density and momentum and $f$ is the Coriolis parameter. The boundary condition

$$U \hat \psi_{zx} - U_z \hat \psi_x = - N^2 U \hat h_x \eqno(17.2)$$

is applied at $z=0$ corresponding to flow over topography. At the upper boundary the condition that waves decay with $z$ (for $\alpha > 0$) is applied.

2: Running the demonstration

On entering this demonstration you must:

(i) specify either a topographic shape and basic state distributions of $U$ and $N^2$, by selecting Specify formulae.

(ii) select Calculate to determine the steady state wave field. This is depicted in terms of the distribution of $\hat \psi$, i.e. of the streamfunction $\psi$ along the centre of the channel, in the $(x , z)$ plane.

(iii) Other menu options control some of the internal parameters of the calculation and are described in detail in §4 below.

3: Notes

The method of solution is to take the Fourier transform of (17.1) with respect to $x$. A set of ordinary differential equations in $z$ 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 $\hat \psi$ as a function of $x$ and $z$.

The method of solution implicitly assumes that the $x$-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 $\alpha$ 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 $\alpha$, $\beta$, $\kappa_h$ and $f$.

Specify formulae: allows you to change the formulae used for $U (z)$, $N^2(z)$ and $h(x)$.

Domain: allows you to specify the ranges of $x$ and $z$ values you are interested in.

Accuracy: allows you to set the number of grid points used in the $x$ and $z$ directions. The number of points in the $x$ 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 $U$ 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 $U$ 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 $N$ increase with height as it does from troposphere to stratosphere, in the real atmosphere. Could the increase in $N$ from troposphere to stratosphere alone account for the observed structure of the waves with height?

References

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.