There is a close qualitative analogy between the quasi-geostrophic system and two-dimensional vortex dynamics (2DVD). It is interesting to consider the quantitative differences in going from one system to the other, associated with the possibility, in the quasi-geostrophic system, of non-trivial vertical structure to the flow. The latter allows inclusion of some extra geophysically relevant physical effects that cannot be represented in 2DVD.

One simple 2DVD system encountered in a previous demonstration is the Kida vortex. This has a quasigeostrophic generalisation in which an ellipsoidal region of constant anomalous potential vorticity evolves under the effect of its own induced circulation, retaining the same ellipsoidal shape, but changing its orientation. If externally imposed horizontal strain and rotation, and vertical shear (i.e. vertical variation of horizontal velocity) is added, the boundary of the region of anomalous potential vorticity remains ellipsoidal, but both its shape and its orientation change. Note that the vertical shear is an ingredient that is beyond the scope of 2DVD, where the flow must be independent of the vertical.

The generalisation to the quasigeostrophic case allows four degrees of freedom for the ellipsoid (an increase of two over the ellipse). Generally an ellipsoid is specified by six degrees of freedom, the lengths of the three semi-axes, plus three angles specifying the orientation of these axes. For the latter it is convenient to use to the so-called Euler angles, conventionally used in rigid-body dynamics. These angles are depicted in Figure 1 (at the end of the documentation for this demonstration. As the ellipsoid evolves there are two constraints on it which reduce the number of actual degrees of freedom to four. The first is that the volume is conserved. The second arises from the fact that the area of the horizontal cross section at each height is constant.

The system is further complicated by the fact that whereas for the Kida vortex there are only two external parameters, the rotation and the strain, for the ellipsoidal vortex there are two further parameters, namely the vertical shear and the angle that the vertical shear makes with the stretching axis of the strain field. Not surprisingly then, the system has not been completely explored. The paper by Meacham et al. (1994) gives a summary of results to date.

1: The model

At each time $t$ the lengths of the semi-axes of the ellipsoid are taken to be $\{ a(t), b(t), c(t) \}$ and the Euler angles defining the orientation of the axes are $\{ \phi(t), \theta (t), \Phi(t) \}$. Rather than use these as working variables to describe the shape of the ellipsoid we follow Meacham (1994) in using the six quadratic moments $m_i$: $i=1, \dots , 6$, where

$$m_i = \mu \int M_i dx dy dz \eqno (16.1)$$

with

$$M_1 = x^2, \hskip 0.5cm M_2 = xy, \hskip 0.5cm M_3 = y^2, \h... ...5cm M_4 = yz, \hskip 0.5cm M_5 = zx, \hskip 0.5cm M_6 = z^2 .$$

The integral is taken over the interior of the ellipsoid and the constant $\mu = 15 / ( 4 \pi a b c )$.

Note that if the three principal axes of the ellipsoid are aligned along the coordinate axes then $m_1 = a^2$, $m_3 = b^2$, $m_6=c^2$ and $m_2 = m_4 = m_5 =0$. Thus, since the $m_i$ are the six independent components of a symmetric second-rank tensor, the lengths of the semi-axes may for general orientations be deduced from the $m_i$s by finding the eigenvalues of that tensor.

The boundary of the ellipsoid is a material surface and its evolution may be followed by calculating the velocity field on the boundary. The important property of an ellipsoidal shaped region of anomalous potential vorticity, as for an elliptical region of anomalous vorticity in 2DVD, is that the velocity field, induced by the potential vorticity anomaly, on the boundary is a linear function of the coordinates. Therefore as the boundary deforms under this velocity field it retains an ellipsoidal shape. Furthermore, this holds if an externally imposed velocity field is added, provided that is a linear function of the coordinates. In this case we take the imposed velocity field to have components $( - \gamma y + e x + \tau_x z , \gamma x - e y + \tau_y z , 0 )$, where $\gamma$, $e$, $\tau_x$ and $\tau_y$ are respectively the vorticity, strain rate and the two components of the vertical shear, i.e. the vertical gradient of the horizontal velocity. Without vertical shear the externally imposed flow is exactly that in the Kida vortex demonstration.

The equations for the time evolution of the $m_i$s may be shown to be

$$\dot m_1 = 2 e m_1 - 2 \gamma m_2 + 2 \tau_x m_5 - 2 V \{ ( m_2 m_6 - m_4 m_5 )I_1 + m_2 I_2 \}, \eqno(16.2a)$$

$$\dot m_2 = \gamma m_1 - \gamma m_3 + \tau_y m_5 + \tau_x m_4... ... ) + ( m_4^2 - m_5^2 )]I_1 + (m_1 -m_3 )I_2 \} , \eqno(16.2b)$$

$$\dot m_3 = 2 \gamma m_2 - 2 e m_3 + 2 \tau_y m_4 + 2 V \{ ( m_2 m_6 - m_4 m_5 ) I_1 + m_2 I_2 \} , \eqno(16.2c)$$

$$\dot m_4 = \gamma m_5 - e m_4 + \tau_y m_6 + V \{ [ m_3 m_5 - m_4 m_2 ] I_1 + m_5 I_2 \}, \eqno(16.2d)$$

$$\dot m_5 = e m_5 - \gamma m_4 + \tau_x m_6 - V \{ [ m_1 m_4 - m_5 m_2 ] I_1 + m_4 I_2 \}, \eqno(16.2e)$$

and

$$\dot m_6 = 0 , \eqno(16.2e)$$

where $V = \Omega a b c$, $\Omega$ being the anomalous potential vorticity in the ellipsoid, and for $j =1,2$

$$I_j = \textstyle {1 \over 2}\int _0^\infty s^j [(a^2 + s) (b^2 + s)(c^2 + s) ]^{-3/2} ds.$$

Clearly $m_6$ is a constant of the motion, corresponding to the second constraint mentioned earlier. It is also the case that the quantity $C = m_1 m_3 m_6 + 2 m_2 m_4 m_5 - m_1 m_4^2 - m_3 m_5^2 - m_6 m_2^2$, which is propotional to the square of the volume of the ellipsoid is a constant of the equations above. Rather than using this constant to eliminate one of the five variable $m_1$, ..., $m_5$ the demonstration integrates the five equations as if they were independent and checks the accuracy of the calculation by monitoring changes in $C$. As the integration proceeds the values of $a$, $b$ and $c$ are calculated and the integrals $I_1$ and $I_2$ integrated numerically.

2: Running the demonstration

On entering this demonstration you must:

(i) Set the values of the parameters $\Omega$, $\gamma$, $e$ and $( \tau_x , \tau_y )$ using the Model, option.

(ii) Choose Initial Conditions for the ellipsoid defining the lengths of the semi-axes $a$, $b$ and $c$ and the initial orientation of the principal axes.

(iii) Set the ellipsoid in motion by selecting Go.

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

3: Notes

The most delicate part of the numerical scheme is the evaluation of the integrals $I_1$ and $I_2$, for different $a$, $b$ and $c$. This can sometimes, particularly when $a$, $b$ and $c$ are several orders of magnitude different, lead to problems, manifested by a substantial change in volume of the ellipsoid. You should also ensure, however, that the time step $\delta t$ is somewhat smaller than the inverse of the largest of $\Omega$, $e$, $\gamma$, $\tau_x$ and $\tau_y$. Decreasing $\delta t$ improves the accuracy and reduces the tendency for volume changes.

4: Menu Options

Go: starts the evolution of the system. You will see the ellipsoid move relative to a fixed set of reference axes. As the ellipsoid changes size, the representation of the reference axes is scaled to a similar size, although the axes and length scales themselves are fixed. The <ESC>key may be used to halt the integration, in which case Go may be selected to resume it again. You can change the angle with which to view the reference axes ( and your distance from them ) whilst the calculation is running, in the same way as for the Viewpoint/Adjust Viewing Angle option described below.

If you have used Flip to select a phase plane display, the evolution will instead proceed on this screen. In this case your only option is to press <ESC>or the left mouse button to end. If the state of the system moves out of bounds for the phase plane, the display will automatically re-scale to make room. If you halt and give a new initial condition before restarting, the new curve will be drawn in a different colour. The program attempts to remember as much of the phase plane as possible between runs, until you choose to clear it.

Initial Conditions: Options to specify the initial shape and orientation of the ellipsoid. Its position remains centred around the origin of the reference axes at all times. Leads to the submenu:

Specify: Pops up a requester in which you can specify the lengths of the three semi-axes $a$, $b$ and $c$ and also their orientation relative to the reference axes. The current values of these parameters are shown. If you do not want to alter them you can simply press <ESC>as usual.

The orientation of the principal axes is specified by the Euler angles $\theta$, $\phi$ and $\Phi$. The position of the principal axes is calculated in the following way (depicted in Figure 1):

3(i): begin with the reference axes and rotate by angle $\phi$ about the $z$-axis.

3(ii): rotate by angle $\theta$ about the new $x$-axis.

3(iii): rotate by angle $\Phi$ about the new $z$-axis.

Adjust: Options which allow you to modify the ellipsoid interactively, i.e. with the perspective window continuously updated to reflect the changes you make. Leads to the submenus Orientation and Shape (see below).

When you select an interactive adjustment, you are presented with the perspective display and a blue legend to indicate the option you selected. The three possibilities are Rotate Ellipsoid, Modify Shape and Rotate Viewpoint. You can switch to one of the other modes without returning to the menu by pressing Enter.

In each case you may press a key to select the axis to be rotated or enlarged, and the result is immediately redisplayed. Pressing <Shift> plus the appropriate key has the opposite effect ( a negative rotation or a contraction ). Adjustments to angles are made in single steps of $\pi / 16$ radians, and enlargements are by 25%.

In all three of the interactive modes, several general options are available. The keys <PgUp> and <PgDn> provide crude control over the viewpoint distance parameters (see below). All three are scaled up or down respectively by a factor of 2. This is useful to perform a quick zoom if your ellipsoid is too big or you can't see the axes. Changes made are retained when you quit the interactive mode.

You can press the <#>key to toggle an alternative method of specifying angle, in which rotations are applied continuously. When continuous rotations are selected, the mode legend is displayed in yellow. The precision available with continuous rotations is $\pi / 48$ radians.

When you are satisfied with your adjustment, press <ESC>to return to the menu.

3Orientation: The blue legend says Rotate Ellipsoid. Keys A, B and C rotate the ellipsoid about one of its own principal axes. Keys X, Y and Z rotate it around one of the reference axes.

3Shape: The blue legend says Modify Shape. Keys A, B and C adjust the length of each principal axis in turn.

Default Shape: The lengths of the principal axes are reset to their defaults. Presently this means $a = 1$, $b = 0.5$, $c = 2$.

Default Orientation: The principal axes are aligned with the reference axes. This corresponds to the Euler angles $\theta$, $\phi$ and $\Phi$ all equal to zero.

Viewpoint: Options to adjust the position of the observer and the perspective window relative to the reference axes ( which are always centred at the origin in 3 dimensional space ). Leads to the submenu:

Adjust Viewing Angle: This is an interactive adjustment which works in the same way as the Initial Conditions/Adjust options. On selecting this submenu the blue legend says Rotate Viewpoint. Keys X, Y and Z rotate the reference axes about themselves ( E.g. pressing X leaves the $x$ reference axis alone and rotates the other two about it ). Keys 1, 2 and 3 rotate the reference axes about a set of notional axes centred at the origin and aligned with the perspective window. Axis 1 is always horizontal, 2 is into the screen, and 3 is vertical.

Default Viewing Angle: Resets the viewing angle, so that two of the reference axes are parallel to the perspective window and the third perpendicular to it.

Distances: You can specify the following:

3(i) the size of the perspective window in the three dimensional world,

3(ii) the perpendicular distance between the perspective window and the origin,

3(iii) the perpendicular distance between the observer and the perspective window.

Reset distances returns the above distances to their default values.

Model: Parameters relating to the specification of the system and the method of numerical solution.

Flow Parameters: Set the values of the rotation rate, $\gamma$, anomalous potential vorticity $\Omega$, strain rate $e$ and vertical shear $( \tau_x , \tau_y )$.

Time step: sets the timestep $\delta t$. Note that the perspective vortex display is only updated every 10 timesteps, to speed up the process. However, the evolution of the system is always calculated using the timestep that you have specified. Note also that the current time display is not continuously updated when you are evolving in phase plane mode.

Rendering: These options determine the way the ellipsoid is presented to you on the perspective display. Only the graphical representation is affected. Leads to the submenu:

Graphics Style: you may specify the Mesh type and Fill type. The Mesh type may be toggled between Globe and Dome to specify the way in which the ellipsoid is tiled. If you select Globe then lines of latitude and longitude are drawn, as on a globe of the Earth's surface. If you select Dome then a tetrahedron is repeatedly subdivided to achieve a tiling with triangles only. This tiling has less discernable structure.

Fill type may be toggled between Lines and Points. If you choose Lines then the ellipsoid is rendered as a polyhedron. If you choose Points then the straight lines describing the edges of the faces are omitted, and you just see the vertices.

Detail Levels: The first parameter, Dome resolution, is the number of times the tetrahedron is subdivided if you are using a Dome mesh. The other two parameters specify how many lines of latitude and longitude there should be in the Globe mesh. Reducing the detail levels will speed up the perspective display.

Phase Plane: Options to control the phase plane displays on the flip screen. You should choose Flip before using these, as they have no effect on the perspective screen.

Set state with mouse: You may specify a new initial condition for the ellipsoid by clicking on a point in phase-space with the mouse. This option requires one of the two phase plane displays to be on-screen. Also it is necessary to use Go before this option so that the phase plane display is not blank. Finally, in order to specify the state uniquely this option will insist on the axis-vertical condition, which is explained under Parametrisation below.

To set the state, point the mouse at the desired location on the phase plane and then click the left mouse button or press Enter. If you instead click the right mouse button the program will automatically start evolving from the new state. You can hit <ESC>to cancel if you decide not to change the state. If the desired initial position is off the edge of the phase plane, you can press the arrow keys to expand the display in the appropriate direction prior to selecting your position.

Clear phase plane: clears the phase plane display.

Parametrisation: Selecting this option cycles through three possibilities for the phase plane display:

3(i) Phase Plane $<xy>$ against $<xx>$,

3(ii) Phase Plane ratio against theta,

3(iii) Poincare Section $<yz>$ against $<xx>$.

Here, $<xy>$ means the volume integral of $xy$ over the ellipsoid. `Ratio' indicates the length ratio of the first two principal axes. `Theta' is the angle between the first principal axis and the first reference axis. Displays (i) and (ii) are true phase planes in 2 dimensions. A sufficient condition for these phase planes to specify the ellipsoid's state uniquely is that the 3rd principal axis remains vertical and invariant. This is true initially and will remain so provided vertical shear is zero. A warning will be displayed if you attempt to evolve case (ii) without maintaining the axis-vertical condition, since the output is then not meaningful.

The Poincare Section display is somewhat different to (i) and (ii), showing points instead of continuous lines. A point is plotted only when $<xy>$ happens to be zero. The section therefore shows a two-dimensional cross-section of the intersection of the phase plane trajectory with the $m_2 = 0$ hypersurface.

Print: Provides hardcopy of the phase plane display.

Flip: alternates between the perspective display of the vortex and a choice of phase plane portraits. Note that the phase plane display is not updated if you are evolving on the perspective screen.

Exit: returns to the GEFD main menu.

5: Suggested experiments

(i) First consider the effect of the finite vertical scale of the ellipsoid on the rotation rate. Start with default parameters and orientation. Then increase the length of the vertical axis of the ellipsoid. You should find that the rotation rate decreases as the length decreases. This is a manifestation of the fact that shallow potential vorticity anomalies have a relatively weak velocity signature, but a large density or temperature signal; deep potential vorticity anomalies vice versa.

(ii) Investigate the stability of the default orientation where the ellipsoid rotates about the vertical axis. Perturb the principal axis slightly away from the vertical. Does the perturbation remain small? You should find that the rotation about the vertical axis is stable unless it is the axis of intermediate length that is vertical. What happens in the unstable case?

(iii) What form does the evolution take when none of the principal axes are initially vertical? The evolution might be considered a superposition of rotation about the vertical and nutation of the axis that is closest to vertical. Is the motion periodic? (Look at the trajectory of the motion in the phase plane.)

(iv) In the Kida vortex demonstration you will have seen how the elliptical vortex can resist the effects of external strain. Verify that this is possible for an ellipsoidal vortex, e.g. with one axis vertical. Can the ellipsoidal vortex resist vertical shear? What is the form of the motion? For the Kida vortex, there is a critical strain rate above which all initial conditions lead to unbounded stretching of the ellipse. For strain rates below the critical value some initial conditions allow the ellipse to remain bounded, but, however small the strain rate, there are always some initial conditions which lead to unbounded growth. Can you see why the latter must be true? Do analogous results hold for vertical shear acting on the ellipsoid? First establish that there are some values of the shear for which some initial conditions lead to bounded evolution.

(v) For certain ranges of values of external strain and rotation, with no vertical shear, there are initial conditions giving nutation (or rocking) of the ellipsoid about a vertical principal axis. Using different initial conditions explore phase space for a given set of values of the external parameters. Setting initial conditions in the phase plane using the mouse is a convenient way to do this. You will find some nutating trajectories and other rotating trajectories. The two sorts of trajectories are divided by a special trajectory known as a separatrix. Now consider the effect of adding weak vertical shear to initial conditions that start close to the separatrix. Alternatively tilt the vertical axis slightly to modify such initial conditions. You may find that the resulting trajectory is a complicated, almost certainly chaotic, blend of nutation and rotation.

The ellipsoidal vortex has been advanced as a simple model of long-lived oceanic subsurface eddies. The above results suggest that the time evolution of such eddies might be complex, even in external flow fields that are steady.

References

Meacham, S. P., 1994: Three-dimensional Hamiltonian vortices. 1993 Woods Hole GFD notes, 322-326.

Meacham, S. P., Pankratov, K. K., Shchepetkin, A. F., Zhmur, V. V., 1994: The interaction of ellipsoidal vortices with background shear flows in a stratified fluid. Dyn. Atmos. Oceans, 21, 167-212.