Two-Dimensional Fluid Dynamics





The equations governing the evolution of a strongly magnetized pure electron system are analogous to those of an ideal 2D fluid; electron density is analogous to fluid vorticity. Therefore, we can study 2D vortex dynamics with pure electron systems in a Malmberg-Penning trap. We generate our electron systems with a photocathode, as opposed to the traditional thermionic sources. The photocathode provides increased control over the initial electron profile, permitting us to perform previously inaccessible fluid-type experiments.

The Fluid Analogy

The equations governing the evolution of a strongly magnetized, pure electron plasma are analogous to those of an ideal 2D fluid. Electron density is analogous to fluid vorticity. This means that a cloumn of electrons is analogous to a fluid vortex. Experimentally, fluid vorticity is difficult to manipulate, whereas electron density is relatively easy.


  • gyroradius << dynamical length scales
    • ignore Larmor gyration
  • axial bounce time << dynamical time scales
    • average along z, plasma is essentially 2D
  • 2D motion determined by E x B drifts
    • ignore collisions, finite length effects, and temperature effects
2D Ideal Fluid 

 2D Electron Plasma

$v \equiv $  


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The Photocathode Trap

the Malmberg-Penning trap

The Malmberg-Penning Trap consists of three coaxial, conducting cylinders contained within a high vacuum chamber. The electron columns are confined radially with a static magnetic field B and axially with electric fields (-V is the confining potential). We create the desired initial 2D electron distribution by projecting the appropriate light image onto a cesium antimonide photocathode and grounding the left cylinder; electrons are emitted only where there is light, and they stream along the magnetic field lines into the central confinement region, preserving their distribution. The electrons are confined by applying a negative electric potential to the left cylinder. The distribution is allowed to evolve for a given time, after which the right cylinder is grounded and the electrons are destructively imaged by streaming them onto a phosphor screen. A charge coupled device (CCD) camera detects the resulting image. The image's intensity is proportional to the electron density, and therefore to the vorticity.
example of photocathode's ability


equipotential injections 

inject = 20 ms 
inject = 40 ms 
inject = 1000 ms 

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Decay of Turbulence


Kelvin-Helmholtz Instability

router / rinner = 0.6 
router / rinner = 0.9 

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Vortex Merger

symmetric merger
D = 1.6 dia < merger distance 
D = 2.0 dia > merger distance 
asymmetric merger
point / extended circulation = 0.15 





Vortex within a Vortex


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Vortex Crystallization

no background... no crystal
t = 0.0 ms 
  t = 100 ms 
with background... crystal!
t = 0.0 ms 
t = 2.5 ms 
t = 100 ms 

Vortex Patterns

N = 7 
N = 19 
N = 37 
N = 61 

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"Negative" Vorticity



Jupiter's Great Red Spot - a hollow vortex

Jupiter's Great Red Spot is thought by some to be a "hollow vortex", meaning that there is a deficiency of vorticity in its center. However, hollow vorticies alone are not stable. Youssef and Marcus have proposed that shear generated by Jupiter's zonal winds can stabilize this otherwise unstable vortex.

We are studying this stabilization using pure electron systems. Below, the black dots are two, strong electron columns that behave as vortices and generate a zonal-type flow. This flow has the appropriate shear to stabilize a hollow "anti-vortex", corresponding to the Great Red Spot, where the white ring with the red center is located; the diminished electron density in this region compared to the blue region behaves as negative vorticity (we cannot trap positrons, which would behave as anti-vortices, along with electrons). In the "blue" rotating frame, the red ring with the white center now corresponds to the Great Red Spot; the presence of the white spot indicates that this vortex is hollow. The result that this system is long lived (over 100 vortex rotation times) indicates that a hollow vortex can indeed be stabilized by a zonal flow.

   Laboratory frame

   "Blue" rotating frame (color scale negated, except for black)