This demonstration concerns the response of a symmetric vortex to weak mechanical and thermal forcing. The problem to be considered is relevant in a number of atmospheric and oceanic contexts, including the zonally-averaged structure of the general circulation of the atmosphere, the dynamics of the Antarctic circumpolar current, and the thermal forcing of atmospheric and oceanic vortices.

1: The model

The set of equations to be considered are those for quasi-geostrophic $x$-independent motion on an $f$-plane. In the atmospheric context, with a compressible atmosphere, they take the form

$$\eqalignno{ {\partial u \over \partial t} - fv &= {\cal F}& (... ... {RT \over H} &= {\partial \phi \over \partial z} & (14.3)\cr }$$

$${\partial T \over \partial t} + w \left \{ {\kappa T_{0} (z) ... ...artial T_{0} \over \partial z} \right \} = {\cal Q}\eqno (14.4)$$

$${\partial v \over \partial y} + e^{z/H} {\partial \over \partial z} (e^{-z/H} w) = 0 \eqno (14.5)$$

Here $x$ and $y$ are horizontal Cartesian co-ordinates ($x$ represents longitude and $y$ represents latitude), $t$ is time and the vertical co-ordinate $z$ is log-pressure, defined by $z = -H \ln p / p_{o}$ where $p$ is pressure and $p_{0}$ a standard reference pressure. $(u, v)$ are the horizontal velocity components, $T$ is the departure of the temperature from a background profile $T_{0}(z) , \phi$ is the geopotential height and $w$ is the vertical velocity in log-pressure co-ordinates (equal to $dz/dt$). ${\cal F}$ and ${\cal Q}$ represent respectively the mechanical and thermal forcing. The constants $R$, and $H$ are respectively the gas constant, and the constant scale height, here taken to be 7km. $\kappa$ is equal to $R/c_{p}$, where $c_{p}$ is the specific heat at constant pressure, and for a perfect gas is approximately equal to $2/7$. $f$ is the local value of the Coriolis parameter and is equal to $2 \Omega \sin \phi$ where $\Omega$ is the rotation rate of the Earth and $\phi$ is the latitude.

The equations (14.1-5) are respectively the $x$ and $y$ components of the momentum equation, the hydrostatic relation, the thermodynamic equation and the mass continuity equation. (For those unfamiliar with meteorological conventions, a convenient property of the vertical log-pressure co-ordinate is that it makes the motion appear incompressible.) Note that (14.1), (14.2) and (14.4) are the appropriate forms of the equations for motion linearised about a state of rest, with the further approximation in (14.2) that the $x$ component of the flow is in geostrophic balance.

The equations (14.1-5) may be combined to eliminate all but one variable - in fact it is useful to define a streamfunction $\chi (y,z)$ such that

$$v = -e^{z/H} {\partial \chi \over \partial z} \ \ \ w = e^{z/H} {\partial \chi \over \partial y} \eqno (14.6)$$

and it follows that $\chi$ satisfies the equation

$${\partial ^{2} \chi \over \partial y^{2}} + {f^{2} \over N^{2... ...er N^{2}T_{0}} {\partial {\cal Q}\over \partial y} \eqno (14.7)$$

where the buoyancy frequency $N$ is defined by

$$N^{2} = g \left ( {\kappa \over H} + {1 \over T_{0}(z)} {dT_{0} \over dz} \right ) \ , \eqno (14.8)$$

$g$ being the acceleration due to gravity.

Note that for a Boussinesq fluid the corresponding equation is almost identical, except that the factors $(gH/RT_{0})$ disappear, the term involving $\partial \chi/\partial z$ disappears and the second term on the right hand side should be replaced by $\partial /\partial y$ of the buoyancy forcing, divided by the vertical gradient of buoyancy. The differences are unimportant if the vertical size of the domain is taken to be less than the scale height $H$ (7 km).

In this demonstration the equation (14.7) is solved for $\chi$, given the distribution of ${\cal F}$ and ${\cal Q}$, in a rectangular domain of finite width and height. The condition $\chi = 0$, corresponding to no normal flow is applied at all boundaries. The forcing functions ${\cal F}$ and ${\cal Q}$ are specified by the amplitudes $a_{{\cal F}}$ and $a_{{\cal Q}}$ and the shape functions $\hat {\cal F}( y , z )$ and $\hat {\cal Q}( y , z )$ such that

$${\cal F}= a_{{\cal F}} \hat {\cal F}( y , z ) \eqno (14.9a)$$

and

$${\cal Q}= a_{{\cal Q}} \hat {\cal Q}( y , z ) . \eqno (14.9b)$$

The motivation for using these forms is that the shape functions $\hat {\cal F}$ and $\hat {\cal Q}$ can be specified once and then the amplitudes $a_{{\cal F}}$ and $a_{{\cal Q}}$ varied, thus altering the relative magnitudes of the thermal and mechanical forcing terms.

Once $\chi$ has been calculated, the $y$ and $z$ velocity components $v$ and $w$ may also be derived, together with the Coriolis torque, $fv$, the acceleration $\partial u / \partial t$, the adiabatic warming $N^{2}T_{0}w/g$ and the warming $\partial T / \partial t$. The units for $\partial u / \partial t \ , {\cal F}$, and $fv$ are taken to be ms$^{-1}$days$^{-1}$. and the units for $\partial T / \partial t \ , {\cal Q}$ and $N^{2}T_{0}w/g$ are taken to be K day$^{-1}$. The units for $v$ and $w$ are respectively ms$^{-1}$ and 10$^{-2}$ ms$^{-1}$. These units are most relevant to problems involving the longitudinally averaged circulation of the atmosphere.

2: Running the demonstration

On entering this demonstration you must:

(i) specify the basic state temperature distribution $T_{0} (z)$ by selecting Temperature, and the thermal and mechanical forcing by selecting Forcing Terms and its submenu, and then perhaps Parameters followed by Amplitudes. Default values are preset so that you may proceed directly to (ii) if you wish.

(ii) select Calculate in order to solve for $\chi$ and associated quantities.

3: Notes

This demonstration uses an overrelaxation technique to solve the elliptic equation (14.7). This is highly robust, but has failed in a very few cases. At present there is also a bug in the part of the demonstration which draws a contour plot. This fails if the function to be contoured is constant and there is then no alternative but to reboot the machine and start again.

4: Menu Options

Calculate: starts calculation

Temperature: defines the basic state temperature profile $T_{0} (z)$, either by Mouse or Formula, or Default which resets the profile to the default.

Parameters: leads to submenu:

Latitude: sets the latitude $\phi$, and thereby changes the value of the Coriolis parameter $f = 2 \Omega \sin \phi$

Domain: sets the depth and half-width of the domain.

Amplitudes: are the factors $a_{{\cal F}}$ and $a_{{\cal Q}}$ that multiply the shape functions defining ${\cal F}$ and ${\cal Q}$.

Forcing Terms: leads to the submenu:

Mechanical Forcing: allowing definition of the mechanical forcing ${\cal F}$ using either Mouse or Formula, or Default which resets the mechanical forcing to the default distribution.

Thermal Forcing: allowing definition of the thermal forcing ${\cal Q}$ using either Mouse or Formula, or Default which resets the thermal forcing to the default distribution.

In both these cases the selection of Mouse leads to the submenu:

Get Points: Define the shape function by placing points with specified values in the domain. Define a value (between -1 and 1) and then place the points with that value, clicking the left mouse button to place the points and the right to conclude. Repeat the selection with a new value, until you are satisfied that the function has been sufficiently closely defined. Now select Draw Contours to interpolate to the grid used for the calculations.

Draw Contours: Interpolates the function to grid and draws contour plot.

Clear: clears all points defining the function.

Flip: after the calculation has been completed the demonstration shows six contour plots - meridional velocity ( ms$^{-1}$), Coriolis torque ( ms$^{-1}$days$^{-1}$), vertical velocity (10$^{-2}$ ms$^{-1}$), adiabatic warming ( K day$^{-1}$) and total warming ( K day$^{-1}$). You may change back to original screen (e.g. to compare with the plots of thermal and mechanical forcing) and then toggle between the two screens, selecting this option. Note that you will automatically be returned to the original screen before updating any parameters or forcing distributions. Remember that with the default parameter settings there is a substantial variation in density over the vertical extent of the domain. The vertical and horizontal velocities will not appear to represent a mass conserving circulation unless this density variation is taken into account.

Print: dumps the screen to a printer. (Make sure that the computer you are using is indeed connected to a printer.)

Exit: leaves the demonstration and returns to the main menu.

5: Suggested experiments

(i) One of the purposes of this demonstration is to show that when a mechanical forcing is applied to a rotating stratified fluid the response appears not simply in the velocity field, but also in the temperature field. Similarly, when a thermal forcing is applied the response appears not only in the temperature field, but also in the velocity field. You should, as a start, test these two statements. What, in the first case, leads to a change in the temperature field? How, in the mechanically forced case with rotation, is the acceleration different from that in the non-rotating case (i.e., that which you would deduce from the mechanical forcing alone).

(ii) For the default parameter settings, you should notice that, even if the forcing is symmetric in the vertical, the response is asymmetric, with the largest values coming above the forcing. What can be leading to this? Check that the asymmetry is much weaker when you decrease the size of the domain. (It is probably best to decrease the height and width in equal ratios).

(iii) Investigate the effect on the response of changing $f$ (by changing the latitude $\phi$, but not to exactly zero) and changing the buoyancy frequency $N$ (by changing the temperature profile $T_{0} (z)$). You should find the response becomes taller when $f/N$ is increased and shallower when $f/N$ is decreased.

You might now like to investigate one or both of the following:

(iv) In sudden stratospheric warmings temperatures in the high-latitude middle stratosphere are observed to increase by several degrees per day. One simple model of such phenomena involves the symmetric motion of the atmosphere just as we are considering here, with a suitable westward force being exerted by the dissipation of upward propagating waves. Assume that the warming is observed at 25km. What can you say about the position at which the force must be exerted (given that it is westward)? What might you expect the response to be well above 25km in high latitudes, and at low latitudes. (In fact cooling in the mesosphere, and in the opposite hemisphere are observed at the time of such warmings). Can you show how a westward force might produce an eastward acceleration at certain locations at the same level (also sometimes observed)?

(v) To what extent (under this model) can an ocean current extending over a considerable depth be forced by either wind stress or surface heat exchange with the atmosphere? Try representing the former with a shallow layer of mechanical forcing near the surface, and the latter by a similar shallow layer of thermal forcing (with, say, cooling at high latitudes and heating at low latitudes). What is the structure of the circulation induced by each of these? (Can you think of an important physical mechanism that this model neglects?)