• G. G. J. Ernst asked a question:
    Have blue jets, red sprites, elves above significant explosive volcanic eruption clouds been seen from the International Space Station ?

    Hello everyone,

    A bit of a forum started about the expectation that blue jets, red sprites and elves should be observable above significant eruption clouds associated with explosive volcanic eruptions. As significant explosive volcanic eruptions, typically those that pose a significant threat to international air traffic each year (VEI 3 events or stronger), with clouds reaching above 10km elevation are numerous, such phenomena above eruption clouds should be common.

    Yet it does not seem that they have been reported yet for volcanic clouds.

    All the reports since 1994 seem to focus on, and concern exclusively thunderstorm clouds.

    Two related questions were asked about this and the answers did not report any observations of these phenomena above volcanic clouds yet.

    However the input from the thunderstorm expert community or atmospheric physicists was minimal.

    I am writing to ask those who are familiar with observations made from the International Space Station, or even better those who have been on ISS missions if they may have witnessed blue jets, red sprites and elves above volcanic eruption clouds.

    I would be grateful for any reply or suggestions.

    With best wishes and kindest regards to all,


  • Mariia Bubniak added an answer:
    Is there anyone who has worked on pulsatile flow through tube of elliptic cross section and handled the complexity of Mathieu functions?

    I am working on pulsatile flow of blood in an artery with slightly elliptic cross section and landed up with Mathieu Functions. The computational complexity of dealing with continued fractions and recurrence relations poses a great threat.  There are some classical work on these lines way back in 1950's but due to poor computational facility at that time, the authors have just provided mathematical solutions without carrying out any computational work to understand the flow characteristics.  Any comments on this? References?

    Mariia Bubniak

    My e-mail

    That is why I wait more detail information about your problem. I hope I can help you

  • Shailendhra Karthikeyan asked a question:
    Physical data for an aqueous solution of cetylpyridinium chloride and sodium salicylate (CPyCl/NaSal) are required. 40: 60 or 60: 40 Any help?

    We are planning to use this viscoelastic fluid in our heat transfer problem.  We are not able to get the physical properties of this fluid.  Can anyone help us with linear mixture formula or nonlinear mixture formula to interpolate these values from the individual values of the two components?

  • G. G. J. Ernst added an answer:
    Has anyone ever documented blue jets, elves or sprites above volcanic eruption clouds ?

    Hello everyone,

    I am re-asking a question I have recently asked as I believe I used the wrong keywords to attract the attention of those who may have the answer.

    In the early to mid 1990s, my research led me to anticipate that blue jets, elves and sprites should be generated above 10km+ explosive eruption clouds many times per year.

    I went to a special session on "blue jets, sprites and elves in the ionosphere above thunderstorms" at AGU San Francisco to ask the above question to experts of this in the atmospheric science community, but no one seemed to have paid attention at the time to the possibility that they may commonly be generated above volcanic eruption clouds.

    Much time has flowed by and I wonder if atmospheric scientists may now have detected such ionospheric phenomena about explosive eruption clouds ?

    If not, I believe studying them above eruption clouds may advance both understanding of explosive eruption physics and of  the link between these ionospheric phenomena and thunderstorms.

    Thanking you in advance for any info or  insights.

    Best wishes to everyone,


    G. G. J. Ernst

    Dear all,

    Just to add a little info and to correct mistakes I made in my previous question on the same subject:

    Blue jets have been documented (from 1994, with reports at AGU SF) above thunderstorms (up to 20km elevation or so) as to extending to typically 50km elevation, together with upward thunderbolts, ie. they are in the main stratospheric it seems.

    Above this, purple tendrils have extended up into red sprites up to about 80-90km elevation (ie. from base of stratosphere into mesosphere and ionosphere).

    Around a cluster of red sprites extending lower tendrils annular green elves, much broader in spatial extent have been documented above and around those sprites at 90-100km elevation.

    This is for meteorological thunderstorms and typically observations made from airplanes. Some of those phenomena are flash-like or discharge-like in their initiation and when they are discharged this is accompanied by radio acoutic signals that can be heard from a recorder on the plane for instance.

    I am still awaiting to hear if anyone may have documented the equivalent phenomena expected above thunderstorm-scale explosive eruptions (which are common) or from larger eruptions ?

    Kind regards to all,


  • Shashank Mishra added an answer:
    Can anyone help with simple MHD simulation setup in Fluent?

    Hello everyone,

    I am trying to simulate a simple pipe flow with conductive fluid under the influence of a constant magnetic field. 
    I went through the MHD module of fluent, but failed to set up the problem correctly. 
    If anyone has done MHD simulations with CFD solvers, please guide me. 


    Shashank Mishra

    Hey Amir 

    Thanks for your comment. 

    My case is simple and I want to simulate something like flow in a backstep channel where magnetic field is acting on the region near the backstep. 
    The objective is to see effect of magnetic field on separation. 

    I am trying to code this in Matlab as of now but if I can run in CFD solvers, I can very easily switch to 3D geometry 

    Thanks again 

  • D. Ruffolo added an answer:
    If the magnetic flux tube is not empty but contains a plasma that provides a pressure, how must that pressure look as a function of radius?

    Assume that a cylindrical magnetic flux tube is embedded in a plasma with   P = 108 Pa. Assume that the field is along the z-axis in a cylindrical coordinate system, that the flux tube has a radius of 1 Mm and that there is no plasma inside the flux tube. What does the magnetic field need to be in order for the system to be stable? If now the magnetic flux tube also is twisted so that the field lines go around the flux tube once for every Mm in z, can the flux tube be stable? If the magnetic flux tube is not empty but contains a plasma that provides a pressure, how must that pressure look as a function of radius?

    D. Ruffolo

    Note that some flux tubes are not in force-free equilibrium.  For example, a flux rope from a coronal mass ejection (a type of solar storm) is not in equilibrium and overexpands relative to the solar wind.  In the interplanetary medium, the dynamics are in some sense dominated more by the initial conditions than by a balance of forces.

  • Philippe-A. Bourdin added an answer:
    How can I fix the electrical field in Fluent MHD model which does not work appropriately?

    I try to model charged nanoparticles moving in two parallel plates with voltage on boundary condition. However, the electrical field setting in MHD model in Fluent does not affect the charged particles at all. 


  • Sergey Victorovich Shchepetov added an answer:
    Any suggestions for the boundary conditions of the magnetic field equation in MHD?
    There is an equation: dB/dt - curl( v x B) = - curl ( L curl B) for the magnetic field B in MagnetoHydroDynamic domains. For the boundary conditions we have curl B = -mu j (j is the electric current density)
    Sergey Victorovich Shchepetov

    Boundary conditions are obtained by integration of equations over a small domain containing boundary. You can find example of such calculation in available book on electrodynamics. In MHD some small terms are neglected.
    Seems you are searching solution with resistive conductors presented somewhere. In this cases it is better to estimate initially whether you conductor thin or thick. If characteristic time of your process is much larger than characteristic skin time of conductor, you need not solve problem on current distribution inside conductor and take it uniform.

  • Daniel Baldomir added an answer:
    Difference between B,H and M in magnetics
    I am somewhat confused with what is B, H and M in magnetism. Can someone explain their equivalence in terms of electrostatics
    Bio Savart law gives us B (which I suppose is magnetic field). But I have read in many places H is magnetics field and is defined as and we have relation as B=mu0*H where B is magnetic flux density. This is somewhat confusing.
    It would be really helpful if someone may explain the following case for the following terms magnetic field, magnetic field strength magnetic flux density and magnetization. Also how mu and mu0 are related in each aspect

    Consider a current carrying conductor in free vacuum. Then at any point Pin free space some B will be felt (I am not giving any name as I am not clear what exactly it is).
    Now suppose the space is not free space but some gas which poses some restriction to flow of magnetic field’s lines through it then if UI am correct, the magnetic field at same pointy will decrease. Is that factor of reduction mu (permeability of gas.
    Daniel Baldomir
    The DOI is:

    DOI 10.1108/03321641111101212

    But it is better that you go to my papers because you can find more things related with your interests.
  • Javier Luis López added an answer:
    Do you have a formula, algorithm, routine c ++, or another programming language that calculates the focus of one or two magnetic lenses?
    I would add it to the next Excel table to upload it to ResearchGate.
    Javier Luis López
    Thank you. I am reading the literature to extract the equations and I downloaded the software.
    My idea is to use two thin lens with different (high) currents to reduce as much as possible the defocusing in the perpendicular and in the axial axle to collide high density ion beams again HLi6 target. It must be impossible to have a little defocusing due dense ions repulsion.
  • John D Sahr added an answer:
    Are solar/space physics in a mature state where all "easy" problems have been done? Are we reduced merely to reducing data and running models?
    Current work in solar physics is dominated by data acquisition, reduction, and numerical modeling. Has the tool between the ears become less important than those catching photons, particles, and those that move electrons around on electrical devices in computers? Is the infrastructure now guiding our research, or are scientific questions guiding the infrastructure we have?

    To see how we might make real progress and avoid some potential traps in relying on external tools, it might be useful to compare thoughts on the major achievements, questions that have been answered in solar physics since, say 1900, and how these were achieved. I might start it off by saying that the development of the Saha equation allowed us to begin to understand solar spectra for the first time, in a quantitative fashion, for example.
    John D Sahr
    A gripping discussion. I'm not sure that I can add to it, other than to say that I am sympathetic to Bo's point of view, and have been for years.

    Space Physics has a number of fundamental obstacles to scientific profundity (as Bo put it). Space plasmas occupy a middle ground --- they're not particularly hot, not particularly cold, not particularly dense (they are the most tenuous). They are resistant to laboratory simulation (e.g. length scales too large for vacuum chambers). They are difficult to study in situ (satellites are expensive, have very long program duration, and often travel at supersonic speeds through the plasma). Space plasmas are difficult to study remotely -- the most useful radar instrument (incoherent scatter) is quite expensive to build and operate. And of course, space plasmas are difficult to simulate computationally because of the range of mass, interaction length, and interaction times involved.

    Yet another challenge is that Space Weather has not risen in the mind of the public to the level of Tropospheric Weather. Everyone knows what it's like to get rain on their picnic, but hardly anyone notices when the Kp index exceeds 7, or when there has been an X30 solar flare.

    These are challenging times for space physicists.
  • Maysam Zamani Pedram added an answer:
    What is the effect of gravity on Magneto-microfluidics?
    A credible amount of literature exists on the magnetic and fluidic forces experienced by a magnetic particle surrounded by a flowing medium in a microfluidic channel when exposed to a constant magnetic field. But the effect of gravity is seldom spoken of. What is the effect of gravity on such a particle despite its low magnitude?
    Maysam Zamani Pedram
    Because most of the researches have been done in variation of parameters in steady states. the force affected by the gravity neutralize by buoyancy force. in some many studies force in planar have been investigated and remained direction has been neglected becuase it does not any effect on the dynamic motion of particle
  • Nishant Narechania added an answer:
    Benchmark Problem for MHD?
    I am looking for a benchmark problem for MagnetoHydroDynamics to validate a numerical code.
    Nishant Narechania
    Hello Amin, you are welcome. But, I do not have knowledge of MHD in molten metals or arc plasma. I would know more about space plasma applications.
  • Janis Priede added an answer:
    What is the relation for entrance length of magnetohydrodynamic flow for rectangular channels?
    I need some help to find out empirical relation for entrance length in channel flow in terms of Hartmann number and Reynolds number. As far as I know it is proportional to Ha/Re. I think the following paper has given some relation about this:

    W. T. SNYDER. "Magnetohydrodynamic flow in the entrance region of a parallel-plate channel." AIAA Journal, Vol. 3.
    Janis Priede
    In that case you could just take a developed flow profile a few 'diameters' upstream from the bend and forget about the entrance length...
  • Vasily Ivanovich Erofeev added an answer:
    What is the advantage of Hamiltonian mechanics in describing transport and turbulence in magnetized plasma?
    I need to know in detail why we prefer to use Hamiltonian mechanics for transport and turbulence in the confined hot plasma rather than using the concept of anomalous diffusion.
    Vasily Ivanovich Erofeev
    Dear Hamdi Abd-El-Hamid, I would like to give you an extra answer to your question.

    As a matter of fact, an appear of hamiltonian approaches to modeling nonlinear plasma phenomena was conditioned by traditions of plasma theory, and the latter were natural to the bed of development of theoretical physics in the lapsed XX-th century. Meanwhile, it is just the plasma theoretical studies that have helped to uncover that the very above traditions do insufficiently well comply with basic goal of physical theoretical researches, namely, with a desire to formulate possibly more informative conclusions regarding physical phenomena in surrounding world. Let me expose you the basic argumentation in support of this statement, and also unroll a bit more its content.

    In the nuclear fusion research, the most natural basis of plasma consideration constitute the full plasma description. It is given by simultaneous Maxwell equations and Klimontovich-Dupree equation: They contain in cumulative form the classical motion equations of all individual charged plasma particles. (I comment that respective plasmas have high temperatures, therefore the notion of particle trajectory is well consistent.) However, the full plasma description is not constructive: its equations cannot be integrated to full extent. (The reason is that, on the one side, the data on initial positions and momentums of individual plasma particles are never known to full extent. On the other, even had it been known, there were not possible to integrate the equations from the technical viewpoint, because of immensely large numbers of charged particles.) In view of this, plasma theorists should reduce the full plasma description to more easy models of plasma kinetics. Just the simplified models of plasma kinetic were underlain, particularly, both the modeling of the nonlinear plasma phenomena by a Hamiltonian and the machinery of theoretical consideration of the anomalous (say, neoclassical) transport phenomena in ionized plasmas. Meanwhile, traditional practice of reducing full plasma description to simplified plasma kinetic models was formed without attention to the most important aspects of the problem. They can be clarified as follows.

    Note that none of constructive plasma descriptions (i.e., the inevitably simplified ones as compared to the full plasma description) can give a scenario of the plasma macrophysical evolution that does not diverge over time from real physical picture of the plasma evolution. (Recall above mentioned incompleteness of initial data of plasma particles and the technical impossibility of its full-scale account.) Meanwhile, the plasma macrophysical evolution constitute the key interest in plasma research. Therefore, an extremely important aspect of a theory constitutes the degree of adequateness of its prognoses on behavior of evolving plasmas to real pictures of their macrophysical evolutions. I introduce there the category INFORMATIVENESS: The longer the theoretical scenario of the plasma evolution depicts the real picture of the macrophysical evolution of the plasma, the higher an estimate the researcher should suggest for the scenario informativeness. Respectively, the heightening of informativeness of plasma scenario to real plasma physical phenomena in surrounding world should constitute the leading motif of the theory development. It supposes an extremely careful separation of the theory informational basis from the full (unknown) plasma initial data. In the latter respect, traditional methods of developing simplified plasma kinetic models were formulated without proper understanding of two key restrictions which resulted in that respective plasma kinetic models happen to possess by an inappropriately low informativeness. To enlighten the latter to a greater degree, it is sufficient to note that the recipes of traditional theory yield equally rigorous justifications to incompatible versions of the same physical phenomenon. For instance, it helps to substantiate both the thesis of conservation of Langmuir wave quanta in a weakly turbulent plasma and the thesis of an intense wave quanta decay with thermalization of Langmuir wave energy via the stochastic plasma electron acceleration.

    Now - the very shortcomings of traditional approaches. First of them consists in culturally conditioned practice of substituting original plasmas by probabilistic plasma ensembles. It was accepted after the success of the gibbsian statistical thermodynamics: Deductions on reciprocal influence of plasma ensemble statistics were commonly believed to comprise objective laws of the plasma macrophysical evolution. In reality, the ensemble substitution generally obscures the picture of the plasma physical evolution. Following to the most elementary common sense, laws of evolution of plasma ensemble statistics essentially depend on the ensemble content and therefore cannot be regarded as objective laws of the plasma physical evolution.

    Second reason of non-informativeness of traditionally developed plasma physical scenarios stems from an absence of adequate understanding of essence and significance of THE ASYMPTOTIC NATURE OF THEORY CONVERGENCE. The researcher inevitably generates some nonlinear successive perturbations in the process of reducing full plasma description to simpler models of plasma macrophysical behavior. Regardless their essence, they may converge at best ASYMPTOTICALLY only: The heightening of the order of consideration entails the factorial growth of number of new summands in the perturbation theory, and after some order the factorial growth of this number outweighs the possible power-law decrease of individual summands. Respectively, one may rigorously deduce diverse pictures of the plasma physical evolution even using a single perturbation theory, having merely varied its lowest order approximation. The point is that the choice of the theory leading order specifies respective conditional limit of converging (first coming) sequential orders of the theory, whereas the latter comprises corresponding version of the plasma physical scenario.

    Two above reasons of theory non-informativeness cannot be separated (without unrolling of the thesis in these my explanations).

    Having formulated the problem of heightening the informativeness of plasma theoretical deductions, I have clarified basic principles for its solving. First, the researcher should refrain from the traditional plasma ensemble substitution.
    Such a refrain forces to modify the concept of basic objects of the theory, THE DISTRIBUTION FUNCTIONS OF PLASMA PARTICLES. Mathematically, it represents some STATISTIC of distribution of discrete charged particles in phase space of particle positions and momentums. Usual approaches implied the
    developing of such a statistic, the distribution function of a Vlasovian type, just via the ensemble averaging of its counterpart from full plasma description (i.e., from the Klimontovich's distribution function of charged plasma particles). The only constructive way to refrain from the ensemble averaging consists in its substitution by a contextually oriented averaging in phase space of positions and momentums of plasma particles. An appropriate arrangement of the averaging depends essentially on the physical problem under consideration. In the case of a homogeneous plasma with weakly turbulent wave fields one can average over 6-dimensional parallelepipeds with extended spatial dimensions: at the expense of these dimensions one can reach appropriate small momentum gradations of a statistically reliable particle distribution. The consideration of plasma leakage from a tokamak dictates toroidal geometry of spatial projection of a 6-dimensional averaging volume. Naturally, other physical situations shall dictate their own geometrical aspects of the phase space averaging.

    Second principle of developing informative plasma kinetic scenarios is that the researcher should develop successive iterations of plasma scenario using A DIRECT TIME INTEGRATION of necessary evolutional equations: It is on this path that he can properly account for an available information on current plasma state and its recent history and simultaneously discriminate an indeterminate information on time remote plasma states.

    Finally, a CRUCIAL position in developing informative models of plasma physical evolution is that its first sequential iterations may result in heightening the accuracy of plasma scenario only when the characteristic expansion parameter of the perturbation theory is small (here I also will not unroll the thesis to a greater extent.)

    The practice of plasma studies consists of many activities whereat the heightening of informativeness of plasma scenarios is strongly desirable. Particularly, just in theoretical studies of plasmas in tokamaks and stellarators that seems to interest you first of all.

    From my above narration understand please that the very question "What is the advantage of Hamiltonian mechanics in describing transport and turbulence in magnetized plasma" is not an important one. Functionally, the hamiltonian approaches substantially simplify considerations of nonlinear plasma phenomena. However, respective simplified considerations have no scientific value, since they cannot lead to appropriately informative pictures of plasma physical evolution. For instance, the hamiltonian formulation of weak plasma turbulence theory does not complies with real macrophysical behavior of weakly turbulent plasmas. Respective truth you may learn only from my papers, ones that were published after 2000. Basically, I have considered a number of phenomena in a homogeneous plasma with weak Langmuir turbulence and have shown that many traditional understandings of respective physics are no more than myths. I have developed for my consideration a new approach that can be well categorized as A HIGH-INFORMATIVE CORRELATION ANALYSIS OF PLASMA KINETICS. My respective studies, and the very technique of plasma kinetic modelling, are exposed in a systematic fashion in my monograph "High-Informative Plasma Theory" (LAP, Saarbrucken, 2011). Naturally, I am performing further on my studies on informative versions of nonlinear phenomena in weakly turbulent plasmas, and above my monograph is a bit outdated. The further extension of my high-informative correlation analysis is reported in new paper, "A High-Informative Version of Nonlinear Transformation of Langmuir Waves to Electromagnetic Waves" (DOI: 10.1017/S002237781300127X, URL = I invite you and other young plasma theorists to learn my ideas and approaches and to adopt them to plasma contexts differing from that of weakly turbulent plasmas. Particularly, this can be said about the problems of transport phenomena. I suspect that consideration of plasma transport phenomena on a basis of new ideas may substantially modify final conclusions.
  • Philip G. Judge added an answer:
    What does happen with the electric characteristics, when a battery is submitted to a high (time variable or not) magnetic field?
    I know, that there are some effects related to magnetohydrodynamics and can change internal resistances, but I'd like to know more about this matter
    Philip G. Judge
    If one considers a wet battery of the old fashioned type as a kind of plasma, with free ions in a background solution, then the addition of slowly varying magnetic fields to the battery will affect the resistance to current flow inside the battery itself, and thereby any external circuit you connect it to. In this case a magnetic field parallel to the direction of normal current flow has no effect. But if you put a magnetic field across this direction it will increase the resistance according to the "magnetization", a parameter (omega*tau) which measures the ion gyrofrequency (omega) times the collision time of the ion with other ions and the solution (tau). The resistance in the direction perpendicular to the magnetic field will go something like (1 + omega*tau)^2. In the liquid state I would expect that tau is very small such that omega*tau is << 1 for most magnetic fields experience on earth. Thus, the magnetic field will have little influence since the ion dynamics is controlled by collisions and not magnetic fields. The clearest explanation of the physics of collisional plasmas is Braginskii's article from 1965 Rev Plasma Phys. 1, p. 205. I hope this helps
  • Andrea Di Vita added an answer:
    Do turbulent magnetic diffusivities in highly conducting plasmas have a sound physical basis?
    In solar physics, it is customary to use magnetic diffusivities which are orders of magnitude larger than kinetic values. But Parker (Space Sci. Rev. 144, 15 2009) points to fundamental difficulties with this concept: the back reaction of the Lorentz force on the plasma frozen to the field on scales above the real diffusion scale can stop the motion of fields through conducting plasma before it reaches the diffusion scale. Under what conditions might a "turbulent diffusivity" be a well grounded concept for highly conducting plasmas, like those in the Sun?
    Andrea Di Vita
    Dear Bian, I have been impressed by your words 'independent of the resistivity '. I wonder if the critical size of physically relevant regions (e.g. near X-points) can be comparable either to ion Larmor size or to collisionless ion skin depth. In both cases MHD fails, and is to replaced by Hall MHD. The relevance of electrical resistivity to the estimate of the current thickness becomes questionable.
  • Abdessamed Medelfef added an answer:
    How about the magnétostrophic approximation?
    I'm working on the dynamo effect action on the Earth's inner core. When we wrote the equations (Navier stokes and induction equation) and we took the asymptotic case we found an equilibrium between the magnetic forces, Coriolis and the gradient of pressure. My question is: is there a unique solution for this case, and where can I find it?
    Abdessamed Medelfef
    Thanx for your help
  • Johannes Gruenwald added an answer:
    In a plasma, is it correct to assume elctrostatics and formulate a debye sphere?
    Plasma contains many charged particles at very high temperatures. So, the different particles produce electro-magnetic fields, which fall off rather slowly compared to the electrostatic fields and more effective. So, is it correct to assume electrostatics and formulate a debye sphere?
    Johannes Gruenwald
    Yes it is. If you can define a Debye length then you can find (per definition) a Debye sphere.
  • Yuriy Voitenko added an answer:
    From frequency to k-space
    If a satellite measures magnetic fluctuations, e.g. in the solar wind, taylors frozen in theorem is usually well satisfied (wave speed << plasma speed) and one can go instantly from freq. to k-space with the transformation k = (2 pi f) / (v sin(psi) ), where psi is the angle between plasma velocity and background magnetic field. So far so good. But how do you determine or at least approximate your trasnformation if taylors frozen in theorem is not that well satisfied? I wonder what happens if the plasma speed vpl is of the same order as alfven speen va (vpl~va). Lets say the fluctuations dB are much smaller than background field B, so that at least dva<
    Yuriy Voitenko
    Whistlers have higher frequencies, so an interference of spatial and time variations is possible (both k and w can contribute to measured f). This should be checked using solar wind parameters in the Doppler-shifted whistler dispersion (which is equal to f).
    BTW I joint the RG network today, just accidentally :)
  • Johannes Gruenwald added an answer:
    Why is Xe in the atmospheres of Earth and Mars much heavier than the solar Xe, but Kr in the atmospheres of the planets is similar to the solar Kr ?
    The noble gas evidence in meteorites.
    Johannes Gruenwald
    Oh, then I misunderstood your question (and again learned something new) - thank you for the paper. :)
  • Charles E Seyler added an answer:
    How to calculate the dependence of the tearing mode 'Delta' parameter on the mode frequency?
    In tearing-mode theory, how to calculate the dependence of the 'Delta' parameter on the mode frequency in the resistive layer?
    Charles E Seyler
    In resistive MHD without equilibrium flow the tearing mode is a purely growing mode and does not have a real frequency. The growth rate (imaginary frequency gamma) depends on delta-prime as gamma^(5/4). If electron inertia dominate resistivity the scaling is different and goes as delta-prime~gamma^(1/2).

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