Dynamically-induced Vertical Pressure Gradient Forces
(Bluestein, Vol. 11, pp 463-468)
In the derivation found on the pages above, the atmosphere is treated as incompressible although density decreases in the vertical (Boussinesq approximation), which simplifies equations of motion. The pressure forces are partitioned into a base state and perturbation from that state (to be discussed in class).
The resulting expressions express the pressure perturbation forces that develop when an initially symmetric rotating updraft develops in a vertically-sheared environment and the impact those have on thunderstorm evolution and behavior. In the equations that follow, p' can be viewed as the perturbation pressure change that would occur at the respective level that would add to or (subtract from) the synoptic scale upwards directed pressure force. In a buoyant updraft, this acts to either add to (or subtract from) the upwards directed buoyancy acceleration.
Term 1: Non-linear Pressure Perturbation
This term says that at a given level surface, pressures will fall (isobars will be found at successively lower elevations) proportional to the square of the relative vertical vorticity. This term is negligible for the kind of vertical vorticity associated with synoptic scale systems. For vertical relative vorticity of order 10-3 s-1 and greater, this term is quite large. Assuming that surface pressures are roughly the same everywhere, decreasing the elevation of the isobars at higher level surfaces (say, the 850 mb isobar) will result in augmented vertical pressure gradient accelerations coincident with the updraft. In essense, this augments the buoyant updraft.
The non-linear pressure perturbation contributes to updraft evolution in both straight (unidirectional) and curved hodograph cases, if the deep layer shear is favorable for supercells. However, it is most clearly demonstrated for a case of supercellular unidirectional shear. Whether the the vorticity that initially develops in the updraft results from tilting streamwise vorticity in the horizontal flow upwards, or (as in the case demonstrated for splitting supercells in Bluestein) from tilting cross-wise vorticity into the vertical by the vertical stretching/deformation of vortex tubes, the initial manifestation is for there to be mirror image updraft augmentation on either flank of the hodograph (of the storm).
The interesting thing about this is that since this process is Galilean invariant (independent of the reference frame), moving the same shear profile around the hodograph produces exactly the same initial storm evolution relative to the hodograph. However, the ground relative motion of the storms could be quite different.
Term 2: Linear Pressure Perturbation
This term says that for a given layer, if the wind shear vector lies across a gradient of the perturbation vertical motion field (e.g., the vertical motion field due to buoyancy), that there will be a pressure perturbation at the level of the layer in question. If the wind shear vector lies across a buoyant updraft, pressures will rise on the upshear side and fall on the downshear side.
This creates an augmented upward pressure gradient acceleration on the right flank of storms growing in a shear environment in which the shear vector veers with height, by lowering the pressures in the lower mid troposphere and raising the pressures in the near surface layer (as shown in the demonstration). "Suppression" of the updraft takes place on the left flank. For the kinds of hodograph often observed in the Great Plains (and in California), the impact of the linear pressure perturbation term is to augment the buoyancy by a factor of 2 to 3 times, creating much stronger updrafts for such supercells than for supercells that merely have the non-linear pressure perturbation force as a dominant effect.
Furthermore, the deviate motion actually creates a situation that favors (a) a longer lived storm; and (b) a storm that is liable to tilt additional horizontal vorticity (generated solenoidally along the forward flank outflow boundary) into the vertical, augmenting the rotation in the mesocyclone. In a perfectly curved hodograph, after the inital split (because of Term 1), the left mover will be suppressed quickly, because of the downward directed pressure forces on that flank of the original updraft.
Of course, in real weather situations, hodographs often show strong curvature in the lowest layers and unidirectional shear above that. Usually, storms evolve conceptually between the two extremes mentioned above (mirror image supercells for perfectly straight hodograph, and a strongly deviate right mover with the original left mover suppressed nearly right away).
Here is a case in point. The hodograph for Amarillo for 12 UTC May 24, 2003 shows a marked loop in the lowest layers, with unidirectional shear above that shallow layer. Conceptually, the expected storm evolution would be a combination of the bottom case shown here and a totally dominant, strongly deviate right moving storm. That's exactly what happened. The right mover moved southwest and the left mover progressed very slowly northward. Two tornadoes occured with the right mover.
In this case, a careful consideration of the hodograph would direct attention to the northwest portion of the cumulonimbus cloud, a region in which the wall cloud would be found, because the entire hodograph is rotated about 90 degrees clockwise from that most commonly observed in the Plains (with southwesterly flow aloft). Chasers expecting to find a wall cloud on the usual southeast flank of the storm would find a shelf there instead.