The equatorial quasi-biennial oscillation is manifested in the zonal wind in the tropical lower stratosphere, between about 18km and 35km, which is observed to undergo an oscillation with average period around 26 months. It is generally accepted that the mechanism for this oscillation is the interaction between vertically propagating waves generated in the tropical troposphere (although planetary waves generated in the extratropics may also play a role) and the longitudinally averaged flow. The waves involved in this phenomena in the real atmosphere almost certainly include equatorial Kelvin waves and equatorial Rossby-gravity waves, and the background rotation of the Earth is an important factor in the behaviour of these waves, as well as in the reponse of the mean flow. However, the basic mechanism can be demonstrated in non-rotating stratified flow in which the mean flow depends only on the vertical coordinate and the waves are internal gravity waves forced at the lower boundary.

1: The model

We consider non-rotating stratified flow in which small-amplitude internal waves are forced, e.g. by the oscillation of a lower boundary. It is assumed that the waves are described by linear theory and that the flow is sufficiently slowly varying in the vertical that the WKBJ approximation may be applied. The wave field is made up of a number $M$ of independent components, the $n$th component having phase speed $c_{n}$ in the $x$-direction. The wave field is assumed to be homogeneous in the $y$ direction and the $y$ coordinate may therefore be dropped. The equation for the $x$-averaged velocity component in the $x$-direction, $u ( z , t )$, where $z$ is the vertical coordinate and $t$ is time, is

$${\partial u \over \partial t} = - \sum_{m=1}^{M} {\partial F... ...} u \over \partial z ^{2}} + \beta ( \bar u - u ) \eqno (18.1)$$

where $F_{n}$ is the vertical momentum flux due to the $n$th wave component, $\nu$ is a vertical diffusivity of momentum and the extra term $\beta ( \bar u ( z ) - u )$ has been included to allow the possibility of a systematic force which acts to bring $u$ back towards a `background' flow profile $\bar u ( z )$. The boundary conditions applied to (18.1) are $u = 0$ at the lower boundary $z = 0$ and $\partial u / \partial z = 0$ at the upper boundary $z = z_{T}$. Insight from simple wave models leads to the restriction that the sign of the momentum flux $F_{n}^{0}$ associated with each wave component at the lower boundary is the same as that of the intrinsic phase speed $c - u$ at the lower boundary.

Applying WKBJ theory and assuming that the momentum flux due to the $n$th wave component is maintained at a constant value $F_{n}^{0}$ at the lower boundary, the variation of $F_{n} ( z , t )$ with height may be represented by

$$F_{n} ( z , t ) = F_{n}^{0} \exp [ - \int_{0}^{z} g_{n} ( z' , t ) dz' ] \eqno (18.2)$$

where, on the assumption that the decay of the waves is primarily due to thermal dissipation, the vertical decay rate $g_{n}$ may be written as

$$g_{n} ( z , t ) = { \alpha \over k_{n} ( u ( z , t ) - c_{n} )^{2}} , \eqno (18.3)$$

$k_{n}$ being the wavenumber in the $x$-direction of the $n$th wave and $\alpha$ being the product of the buoyancy frequency and the thermal damping. Thus, given the constants $F_{n}^{0}$, $k_{n}$, $c_{n}$ (for $1 \le n \le M$) and $\alpha$, (18.3) may be substituted into (18.2) and then into (18.1) to give a closed partial differential equation for $u ( z , t )$.

The main physical mechanism leading to (18.3) is that as $u ( z , t ) - c$ decreases the vertical group velocity decreases and therefore the spatial decay rate due to thermal dissipation increases, so strongly that if $u ( z , t ) - c$ changes sign the wave completely disappears. This is a qualitatively valid representation of the behaviour of realistic internal gravity waves on shear flows provided that dissipation is sufficiently strong.

2: Running the demonstration

On entering this demonstration you may:

(i) start the time evolution of the flow by selecting Go. The evolution may be halted by pressing <ESC> and then resumed by selecting Go again. Default profiles of the initial flow $u(z,0)$ and the background flow $\bar u ( z )$, i.e. that to which $u ( z , t )$ is relaxed at rate $\beta$, (if $\beta > 0$), are set up for you when the program is started. The default for the latter profile, which for most purposes is all that you will need, is $\bar u ( z ) = 0$ (and also $\beta = 0$).

(ii) Should you wish to change either initial or background flow profiles then you must first select Clear. Both profiles must then be redefined, either via the mouse or by selecting Get default. Both are constrained to be zero at $z = 0$.

(iii) Should you wish to change the parameters of the model, in particular those defining the wave field, you should select Change Parameters. More details are given in §4, but an important requirement to be satisfied is that if the number of waves $M$ is increased, the corresponding wavenumbers and phase speeds must be non zero.

3: Notes

The scheme used to integrate the model equations is fairly robust, but as usual it is required that the time step be fairly small in the appropriate sense. It is therefore advisable that the lower boundary momentum fluxes $F_{n}^{0}$ have amplitudes less than or equal to 1 and that the time step $\Delta t$ satisfies $\Delta t\textstyle {< \atop \sim}0.2$.

The diffusivity $\nu$ plays an important physical role in this model, but it also acts to smooth out sharp features in the profile of $u ( z , t )$ associated with the finite vertical grid size $\Delta z$ and you should therefore not be too surprised to see such features appear if $\nu$ is given a very small value ( e.g. less than $0.05$ ).

4: Menu Options

Go: starts the time evolution.

Get u(z): defines the initial flow profile $u(z,0)$ (constrained to $u = 0$ at $z = 0$), using the mouse.

Get ${\tt\bar u(z)}$: defines the profile towards which the system is relaxed. Select Use Mouse to define using the mouse or Get default to set $\bar u = 0$.

Change Parameters: leads to the submenu:

Miscellaneous: defines the time step $\delta t$, the vertical grid size $\delta z$, the diffusivity $\nu$, the number of waves $M$ (maximum 5), the wave damping constant $\alpha$ and the relaxation rate for the mean flow, $\beta$.

Wave numbers: defines the wavenumbers $k_{n}$, for $1 \le n \le 5$. Note that the condition $k_{n} > 0$ for $1 \le n \le M$ must be satisfied.

Phase speeds: defines the phase speeds $c_{n}$, for $1 \le n \le 5$. Note that the condition $c_{n} \ne 0$ for $1 \le n \le M$ must be satisfied.

Momentum fluxes: defines the wave momentum fluxes Fn ($= F_{n}^{0}$), for $1 \le n \le 5$. Note that $F_{n}^{0}$ and $c_{n}$ must have the same sign for each $n$ , $1 \le n \le M$.

Clear: allows redefinition of $u(z,0)$ and $\bar u ( z )$, but leaves other parameters unchanged.

Print: dumps the graphs to a printer. (Make sure that the machine that you are using is connected to a printer.)

5: Suggested experiments

(i) For most purposes it is suggested that you take $\bar u ( z ) = 0$ and $\beta = 0$ (the default values). You should first note that you can use this demonstration to show you the monentum flux pattern for any flow $u ( z )$, without considering the time evolution. Whenever you define $u ( z )$, the corresponding pattern of momentum fluxes will be recalculated. So try a number of different flow profiles and make sure you understand the relationship between the shape of the profile and the variation of the momentum flux with height. For example, take $M = 1$ and $c_{1} = 1$. Try, in succession, profiles with $u = 0$, $u > 0$ and $u < 0$. For which of these choices does the momentum flux decrease most rapidly with height? What would be the corresponding force exterted on the flow? How would the results change if $c_{1} = -1$? (Don't forget that you will have to change the sign of $F_{1}^{0}$.) Now try profiles of $u$ that are positive in the lower half of the domain and negative in the upper half, for $c_{1} > 0$ and $c_{1} < 0$. You might also like to change the damping constant $\alpha$ and note the effect that this has on the monentum flux profile.

(ii) Now consider the time evolution, again with one wave only ($M = 1$). Try a variety of initial conditions, with the default value of the diffusion. You should find that the system reaches a steady state that is independent of the initial conditions. What is the balance of the forces in this steady state? Decrease the value of the diffusivity $\nu$ and note how the steady state changes. Also try changing the magnitudes and signs of the phase speed $c_{1}$ and the momentum flux $F_{1}^{0}$. (Vary the magnitudes of these two quantities independently.)

(iii) Now consider the case studied by Plumb (1977) using the default values of the parameters. There are two waves of equal and opposite phase speeds and momentum fluxes. You should be able to see that $u = 0$ is a possible steady state of this system. (Check that it is.) For the default values of the diffusion you should find that almost all initial choices of $u$ relax to this state. However, if the diffusion is decreased sufficiently you will find that the steady state is not achieved. Try an initial condition that is as close as possible to the possible steady state $u = 0$. What happens? You should find that the flow is unstable to small perturbations, which grow to give a finite amplitude oscillation.

(iv) It is important to understand what controls the amplitude and frequency of the oscillation in the real atmosphere. Find a case which shows oscillations and then try varying the amplitude of the momentum fluxes. What feature of the oscillation changes? Look at the magnitude and the frequency of the oscillation. (You should find that, for sufficiently large values of the momentum fluxes, the frequency changes but not the magnitude.) How can you affect the amplitude of the oscillation? (Hint: Try reducing the phase speeds of the two waves by the same factor.)

(v) You may also like to investigate the behaviour of the system when the waves are not symmetric. There are clearly many possibilities. You might try changing one of the phase speeds, or one of the momentum fluxes. You might also add more waves and see whether you can find oscillations. (It would be worrying if the addition of more waves to the system removed the oscillations, since there are certainly waves of more than two different phase speeds in the tropical atmosphere.)

(vi) The possibility of including a relaxational force $- \beta ( u - \bar u )$ has been added in order to allow investigation of the possibility of oscillations with only one wave in the system. [ An example of a different system that exhibits oscillations under these circumstances is given by Yoden (1988). ]

Note that the model considered here neglects the variation of density with height. Inclusion of such variation leads to changes in the vertical structure with height (Plumb 1977, §7), in particular giving the possibility that the amplitude and phase of the oscillation are almost independent of height sufficiently far above the lower boundary. Whilst the amplitude of the observed oscillation is approximately independent of height above about 23km the phase is certainly not independent of height. Thus adding density variation alone to this model does not lead to complete agreement with the vertical structure of the observed oscillation.

References

Plumb, R. A., 1977: The interaction of two internal waves with the mean flow: implications for the theory of the quasi-biennial oscillation. J. Atmos. Sci., 34, 1847-1858.

Yoden, S., 1988: Bifurcation properties of a stratospheric vacillation model. J. Atmos. Sci., 44, 1723-1733.