ISENTROPIC ERTEL POTENTIAL VORTICITY MAPS ON JUPITER?

In these pages, we consider the possibility of obtaining coarse-grain maps of Ertel potential vorticity on isentropic surfaces (IPV) from observations of temperature and wind velocity in the upper troposphere and stratosphere of Jupiter or Saturn. The overall intention is to use observations obtained during the North-South mapping sequences from Cassini/CIRS together with wind information which might be obtainable from the ISS imaging team.

In the following, we outline the basic theory and methodology, and present a first attempt at deriving approximate maps of IPV from available Voyager IRIS and cloud-tracked wind observations, obtained during the Voyager 1 closest approach to Jupiter in 1979.

A draft paper (in .ps) describing this work in greater detail can be found here. "Isentropic Ertel potential vorticity maps on Jupiter from Voyager 1 IRIS and imaging data"

BASIC THEORY

• For large-scale, hydrostatic flow

In this sub-section we consider the practical methods needed to derive approximate IPV maps for Jupiter itself, adapting the practice of terrestrial meteorologists, developed during the 1980s for the Earth’s stratosphere, to the most likely data sources obtainable from Jupiter orbit or fly-by. In particular, we will not have the benefit of a reliable absolute analysis of winds at a well-defined pressure surface (which terrestrial meteorologists derive from routine operational analyses in the troposphere). The only realistic alternative would seem to be to derive a wind field from tracking cloud tracers, assuming them to lie at or around a roughly known pressure level of ~500 mb. This may well be reasonable, since thermal observations suggest that horizontal temperature gradients (and hence thermal wind shears) are weak in the upper troposphere of Jupiter, and hence the precise pressure level assumed may not be critical to the analysis. Hence the sequence of steps would be

• Form Q from (*)

CASSINI N-S Maps of Jupiter….?

CIRS is planning to make several ~20 hour sequences of maps during closest approach to Jupiter. During this period, the CIRS FP3 and FP4 sensors will slew slowly across the planet N-S, synchronised to enable a complete and (in several cases, fully sampled) global map to be obtained, from which a fully 3D temperature field will be retrieved.

Movement of CIRS FOV during NS scan

• Repeat scan every 25 degs in long. to produce contiguous map

AN ANALYSIS OF VOYAGER DATA FOR THE MEAN ZONAL FLOW

In the following, we try out this procedure using a combination of wind velocity observations, obtained from Voyager imaging data, and Voyager 1 IRIS temperature retrievals from the period around closest approach during the Voyager 1 encounter. IRIS retrievals were kindly provided by Peter Gierasch in the form of temperature profiles and code to interpolate onto a regular grid, and represent the latest version of temperature retrievals obtained from the closest approach dataset (ref?). Zonal mean velocity measurements were kindly provided by Amy Simon-Miller, and were smoothed to give a nominal resolution in latitude of about 2 degrees.

Frames showing the smoothed zonal velocity data, and profiles of the relative vorticity gradient with the usual planetary vorticity gradient (beta) shown dashed. This clearly shows the apparent reversals in the sign of beta - u_yy at various latitudes.

From the temperature field, we can obtain the horizontal thermal gradients, and hence integrate the thermal wind equation upwards from the wind field at the lower boundary. In practice we integrate the wind field directly, and then form the vorticity at each pressure level. We show below a latitude-pressure section of the zonal wind field derived by this method. Note that the winds appear to decay with height in the southern hemisphere, but are apparently more barotropic in the northern hemisphere...?

I have implemented a new definition of potential temperature discussed by Peter Gierasch, which takes account of the fact that specific entropy of a hydrogen/helium mixture (and hence potential temperature) depends not only on T and p, but also on the para-hydrogen fraction fp, which measurements suggest is variable in latitude. The new definition of potential temperature is designed to converge towards the more traditional (single-component fluid) definition at both very high and very low temperatures, and departs only a little (by 5-10K) from this at intermediate temperatures (for more details, see below on the GRS).

From the 2D fields of relative vorticity and potential temperature, we form the approximate Ertel PV (using the ABSOLUTE vorticity, adding in the planetary component f), and interpolate onto theta-surfaces. The 'final' result (now using the new version of potential temperature) is shown below as a cross-section in latitude-potential temperature. Note the change of sign of PV across the equator (the equatorial region itself is clear, because we cannot compute thermal winds too close to the equator) and the near-steplike form of the contours. Some apparent 'glitches' occur at certain latitudes, especially close to the equatorial zone and the 21 degs north jet.

Finally, we show below a couple of latitudinal profiles of IPV on selected isentropic surfaces - 180K (around 400mb) and 250K (above the tropopause). This does again illustrate structure suggestive of steps, with sharp gradients near westerly jets......?

AN ANALYSIS OF VOYAGER DATA FOR THE GRS

In the following, we try out this procedure using a combination of wind velocity observations, obtained from Voyager imaging data, and Voyager 1 IRIS temperature retrievals from the period around closest approach during the Voyager 1 encounter. IRIS retrievals were kindly provided by Peter Gierasch in the form of temperature profiles and code to interpolate onto a regular long. x lat. x pressure grid, and represent the latest version of temperature retrievals obtained from the closest approach dataset (ref?). Velocity measurements were kindly provided by Tim Dowling, and consist of a series of around 2200 individual velocity measurements derived from the displacement of cloud tracers in image pairs, ~ randomly distributed around the region of the GRS. The data were actually combined from both Voyager 1 AND Voyager 2 images by Dowling & Ingersoll (1988, JAS, 45, 1380-1396), and have been shifted in longitude by 31.48 degrees from the coordinate frame published by Dowling & Ingersoll to bring them into alignment with the location of the GRS at the time of the Voyager 1 encounter.

A frame showing the raw velocity data superimposed on an IRIS temperature map at p ~ 120mb is shown below.

The velocity field was interpolated by IDL onto a regular 1 x 1 degree grid in (long. x lat.) [I'm grateful to Steve Lewis at Oxford for pointing out how to do this elegantly!]

From this, fields of relative vorticity and divergence could be obtained, the former of which forms the basis for the lower boundary condition for the computation of IPV. Note that the wind field seems close to non-divergence…..(see RH frame)

From the temperature field, we can obtain the horizontal thermal gradients, and hence integrate the thermal wind equation upwards from the wind field at the lower boundary. In practice we integrate the wind field directly, and then form the vorticity at each pressure level. We show a set of relative vorticity maps below, upon which is superimposed velocity vectors representing the inferred wind field at each level.

In my earlier attempt (November 200), potential temperature was derived fairly naively from the temperature field, using a reference pressure of 1 bar and cp/R = 3.3 (and R = R0/2.23, where R0 is the universal gas constant). Most recently, I have implemented a new definition of potential temperature discussed by Peter Gierasch, which takes account of the fact that specific entropy of a hydrogen/helium mixture (and hence potential temperature) depends not only on T and p, but also on the para-hydrogen fraction fp, which measurements suggest is variable around the GRS. The new definition of potential temperature is designed to converge towards the more traditional (single-component fluid) definition at both very high and very low temperatures, and departs only a little (by 5-10K) from this at intermediate temperatures. The horizontally averaged profiles of potential temperature are shown below, with the old definition on the left and the new definition on the right.

Old pot. temp.	New pot. temp.

From the 3D fields of relative vorticity and potential temperature, we form the approximate Ertel PV (using the ABSOLUTE vorticity, adding in the planetary component f), and interpolate onto theta-surfaces. The 'final' result (now using the new version of potential temperature) is shown below on a series of theta-surfaces, spanning the range in pressure from near the cloud-tops (191K - around 225mb)) to around 10-20mb (450K). Fields show the IPV contours with thermal winds superimposed.

Isentropic PV	fp on theta surface

Fields are still somewhat noisy and 'lumpy', which may reflect the irregular sampling (especially in temperature), and also perhaps the somewhat uncomfortable combination of V1 and V2 velocity vectors in the baseline velocity field? The White Oval just visible in the velocity field south of the GRS, for example, drifted substantially relative to the GRS between the V1 and V2 encounters, and may not have been in the place shown at the time of V1……! However, it does seem to show clearly the GRS as an isolated patch of IPV below the tropopause, becoming an open undulation deflecting the flow around it in the stratosphere.....? There is a hint, also, of filaments of low and high IPV fluid being drawn northwards or southwards around the periphery of the spot.

The relationship between IPV and a suitably-defined streamfunction for the horizontal flow is another valuable diagnostic which could be computed from data. For a steady, almost frictionless, adiabatic flow which is close to ‘free-mode’ form, q =; q(Y). Furthermore, the form of this function may provide information either on the stability of the flow (e.g. via Arnol’d’s second stability theorem) and/or on the integrated PV forcing within regions enclosed by closed streamlines.

The difficulty with this in practice is that z is unknown, even in relative terms, relative to the geopotential on Jupiter. Alternatively, given measurements of horizontal velocity and �q/�p, we can form a streamfunction directly by numerical solution of a Poisson equation. For steady, adiabatic flow on an isentropic surface, the continuity equation becomes

—_q.(u_q dp/dq) = 0

where the subscripts q imply a horizontal divergence on an isentropic surface. Thus, the isentropic streamfunction can be determined from a solution of

—_q²Y = —_qx(u dp/d q).

Given simultaneous fields of Y and q, we can plot a scatter-diagram of one field against the other around the GRS, and this is shown below in preliminary form. The region inside the GRS gives rise to the left-hand branch of the pattern (the red line connects points E-W across the centre of the GRS) showing a near-functional relation between Y and q. Further analysis of these diagnostics is ongoing…….

Hopefully we can do better in all these respects from Cassini - which should give more even, regularly-spaced coverage in T.....