Talk:PlasmaSimulation
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Stability and Averaging
Averaging and Stability
If one constructs a grid in curvilinear coordinates, there may be a variation in mesh size that makes no useful contribution to the accuracy of a hosted calculation, but constrains the time step to a much smaller value than would otherwise be necessary. An example is the coordinate singularity that occurs with polar zoning, and there are many references in global weather modeling to the small zones near the poles, most of them suggesting ways to avoid the singularity (cf Chungan Chen and Feng Xiao, Shallow water model in a cubed sphere by multi-moment finite volume method, J. Comput. Phys. 227 (2008) 5019-5044.)
In an attached pdf file File:Averaging.pdf, we discuss two strategies for easing the time step constraint by averaging the flux in a conservation equation. The analysis is by no means complete. For example, it is not clear why simple cell averaging leads to exponential error growth. Further, it is not clear how averaging using a b-spline in a limited portion of the domain will affect stability and accuracy.
M-I Coupling
What are the appropriate boundary conditions at the interface between a region described by the ideal MHD equations, and one governed by current conservation in a conducting medium? In File:Magnetosphere-Ionosphere Coupling.pdf, the conditions that follow from continuity of the electric field and charge conservation are reviewed.
Slava's comments 5/28/08:
The LFM inner boundary condition on the velocity is a hard wall indeed, so V(+) = V(-) - 2*Vnormal, where V(+) is the velocity just inside of the boundary and V(-) is the velocity just outside of the boundary (in ghost cells). This basically forces Vnormal=0 at the boundary. Vtangential is determined from the ionospheric potential solution.The other variables (including the tangential component of the magnetic field) are continuous across the boundary. The continuity of the tangential magnetic field ensures that the perpendicular (to the field) current is zero at high latitudes where the field is normal to the boundary. At low latitudes this condition is not applicable but is still used and results in noisy field-aligned currents.
Jerry's comments 5/8/08:
My understanding is that magnetic field lines are equipotentials, and thus the magnetic field maps
the potential from the ionosphere to the magnetosphere. Even with the gap, magnetic field lines
connect the ionosphere and magnetosphere electrically. A discontinuity in the tangential electric field
leads to a singularity in 
. (Apply Stoke's theorem to Faraday's law.)
Since 
is undetermined by continuity of the tangential electric field, the 'hard wall' boundary condition, 
is consistent but incomplete.
There is also a tangential flow condition. In Zhang et al., "The electric potential,
which is computed on the 2-D spherical grid of IE [ionosphere electrodynamics module], is mapped back to the inner boundary of GM [global magnetosphere module] (at 2.5 RE in this study) and used to calculate the electric field and velocity at the inner boundary." [p4/14]
The low estimates for the flow velocity in Wolf's 1983 paper are presumably correct. However, 'the high-latitude boundary condition [for the ionospheric potential] is a Dirichlet boundary condition where the solar wind potential is specified as a function of local time' (Toffoletto, 2003, p5). Prolss in "Physics of the Earth's Space Environment" comments that the 'polar cap potential rapidly increases to values many times larger than those observed during quiet times' [p353]. This means there should be anti-solar motion of the polar cap in response.
Slava Merkin's comments:
1. It looks like you have an interface between the magnetosphere and the ionosphere, but this is not quite right -- there is a gap region in between. In particular, J does not equal 
above or below the boundary; the plasma is collisionless. It becomes true below the F-layer which is way below the inner boundary of the MHD domain. The "standard" way is to map the current to the ionosphere assuming 
in the gap region, isn't it?
2. Why does the tangential electric field component have to be constant across the boundary? Does not it prohibit variations in the tangential magnetic field and therefore the field-aligned current?
3. The normal component of the velocity, at least on the equator, is determined by the low-latitude boundary condition applied to the potential solution in the ionosphere.
Slava's comments 4/18/08:
Jerry, see this paper by Bill Lotko (very insightful, in my mind) for some discussion of the gap region physics [1]
The LFM MHD inner boundary condition on the velocity is a hardwall, i.e. . The normal component of the magnetic field is kept constant. Some of this is discussed in John's paper: [2]
Comments 4/17/08:
Is anyone familiar with the book 'Physics of the Earth's space environment: an introduction' by Gerd W. Prolss, M. K. Bird ? I call your attention to Fig. 2.13 on p. 32. Does this remind anyone of the blind men and the elephant?
The inner boundary of typical MHD calculations at 2.5 − 3.0RE coincides with the plasmapause, where the density decreases with altitude by 102 over a distance of 1 / 2RE according to a diagram at onspaceweb at oulu. However the Southwest Research Institute website says:
'The Earth's plasmasphere is a torus of cold (~1 eV), dense (tens to thousands of particles per cubic centimeter) plasma that occupies roughly the same region of the inner magnetosphere as the ring current and radiation belts (between RE = 2andRE = 7) and that is populated by the outflow of ionospheric plasma along mid- and low-latitude magnetic field lines (i. e., those that map to magnetic latitudes of ~60 degrees and less). H+ is the principal plasmaspheric ion, with singly ionized helium accounting for ~20% of the plasmaspheric plasma.'
This suggests that at low latitudes, the plasmasphere will overlap the MHD domain, but that at high latitudes, the magnetic field lines will not traverse the plasmasphere on their way to the ionosphere.
Jerry's comments:
A recent paper says the inner boundary of the LFM grid is an earth-centered sphere with radius 2RE and minimum grid spacing ΔR = 0.3RE, and that field aligned currents map along dipole field lines to a 2D electrostatic ionosphere model. (Huang, C.-L., H. E. Spence, J. G. Lyon, F. R. Toffoletto, H. J. Singer, and S. Sazykin (2006), Storm-time configuration of the inner magnetosphere: Lyon-Fedder-Mobarry MHD code, Tsyganenko model, and GOES observations, J. Geophys. Res., 111, A11S16, doi:10.1029/2006JA011626.) The top of the F-layer, the uppermost layer of the Ionosphere is located between 120 and 400 km above the Earth's surface. The gap that Slava mentions is presumably the interval between the top of the ionosphere and the bottom of the magnetosphere. Perhaps I haven't read carefully enough, but I am unable to determine what are the boundary conditions applied at the inner boundary of the LFM grid.
The Michigan group reported results with coupling to RCM using a grid with inner radius 2.5RE and minimum grid spacing ΔR = 0.25RE (Zhang, J., et al. (2007), Understanding storm-time ring current development through data-model comparisons of a moderate storm, J. Geophys. Res., 112, A04208, doi:10.1029/2006JA011846.) They describe the RCM model as residing in the region 7 − 10RE, but do not give the resolution in that region of the Batsrus grid. Nevertheless, it seems as though the resolution is similar to that in current LFM/RCM calculations. Is that a correct impression? Would more resolution be needed with LFM/RCM than with Batsrus/RCM to do a comparable ring current study?
The LFM paper above describes the application of the Boris correction to limit the maximum signal speed to 1500 − 3000km / s, and the Michigan paper says that the inner radius of their grid is at 2.5RE to avoid the strong magnetic fields near the Earth. Is there any reference which characterizes the variation of the Alfven speed with altitude in the lower magnetosphere? The density presumably decreases exponentially with altitude, the magnetic field intensity decreases as a power of the altitude, so one would expect the Alfven speed to increase with increasing altitude rather than the reverse.
Frank Toffoletto's comments:
I guess I am a bit confused about what it is stating. All the RCM returns to the LFM is mass density and pressure (actually sound speed) where it nudges the LFM values on fieldlines that map to the RCM grid towards values consistent with what the RCM says it should be. The RCM grid is a 2d grid in the ionosphere, all calculations in the RCM are 2d. Conversion to MHD is done via fieldline mapping. No rcm computed velocities are returned to the LFM. The RCM in return gets magnetic field information, plasma boundary conditions end elecrtic field. What happens in the LFM with the RCM nudging it is when the pressure gradients in the inner magnetosphere get large, the magnetic field stretches going from dipolar to tail-like. Eventually high speed flows show up in the LFM and the solution starts to get very messy. We suspect it is poor resolution in the LFM (The radial extent of the entire ring current sits on 3 grid points in the LFM), rather than computing an equilibria where jxB=grad P, we get flows. That is why there has been a push to increase the LFM's resolution.
Slava Merkin's comments:
1. It looks like you have an interface between the magnetosphere and the ionosphere, but this is not quite right -- there is a gap region in between. In particular, J does not equal above or below the boundary; the plasma is collisionless. It becomes true below the F-layer which is way below the inner boundary of the MHD domain. The "standard" way is to map the current to the ionosphere assuming in the gap region, isn't it?
2. Why does the tangential electric field component have to be constant across the boundary? Does not it prohibit variations in the tangential magnetic field and therefore the field-aligned current?
3. The normal component of the velocity, at least on the equator, is determined by the low-latitude boundary condition applied to the potential solution in the ionosphere.
