# What's the difference between “Ohmic dissipation”, “Joule heating”, “ion drag” and “resistive heating”?

The following terms are sometimes used to refer to ... more or less ... the same thing by different people and in different contexts (electronic circuits vs. plasma physics, etc.):

• Ohmic dissipation
• Joule heating
• Ion drag
• Resistive heating
• [other terms??]

I've heard that some of these terms aren't quite the same, or are potentially misleading. See, for instance, the discussion in the following article:

Vasyliūnas, V. M., and P. Song (2005), Meaning of ionospheric Joule heating, J. Geophys. Res., 110, A02301, doi:10.1029/2004JA010615. (link)

What do each of these terms properly or formally refer to, and what are some "best practices" as to when to use one term over the other? What is the best context for each term? Is there a difference in the nature of energy transfer in each case? Which of the various MHD equations would each term show up in?

Note: I'm looking to split hairs, so "they're all the same" isn't a good answer. :-)

• Note: I have accepted the answer of @honeste_vivere, but I am still interested in other's opinions about this. – jvriesem May 4 '15 at 22:45

Joule heating is typically associated with increases in random kinetic energy (i.e., heat) due to $\mathbf{j} \cdot \mathbf{E}$. Ohmic dissipation and resistive heating are similar in a sense to Joule heating, as all three result from fluctuating electric fields acting as an effective drag force on an otherwise free flowing charged particle.

Ion drag is likely associated with fluid terms like viscosity or Coulomb collisions, which can act to inhibit the bulk flow of charged particles.

Generally in a plasma, one refers to anomalous resistivity or anomalous viscosity. The use of the word anomalous comes from the fact that the interactions are not rigorously fluid-like (i.e., not from collisions). They are typically the result from a wave or fluctuating fields radiated by an instability that then act to remove the free energy that created them. Note that I am not referring to the "fudge factor" that MHD simulations will often use to account for or introduce some form of dissipation.

These terms are used in MHD, though in a plasma we have found through observation and particle-in-cell (PIC) simulations that resistive/drag terms arise from purely kinetic effects. Meaning, to have these effects in MHD is to artificially insert them (i.e., throw in some adaptive anomalous resistivity or allow numerical resistivity).

I will edit this answer later with a more thorough response, but my newborn just woke up and needs attention.

The $\mathbf{j} \cdot \mathbf{E}$ comes from Poynting's theorem where: $$\partial_{t} W_{EM} + \nabla \cdot \mathbf{S} = - \mathbf{j} \cdot \mathbf{E}$$ where $W_{EM}$ is the electromagnetic energy density (= $\varepsilon_{o} E^{2}/2$ + $B^{2}/(2 \ \mu_{o})$) and $\mathbf{S}$ is the Poynting flux (= $\mathbf{E} \times \mathbf{B}/\mu_{o}$). Another way to say this is the time rate of change of the energy density of the electromagnetic fields plus the rate of electromagnetic energy flux flowing out of a surface equals the energy lost due to momentum transfer between particles and fields.

When you can approximate $\mathbf{E}$ as $\overleftrightarrow{\eta} \cdot \mathbf{j}$, where $\overleftrightarrow{\eta}$ is a resistivity tensor, then we have: $$\mathbf{j} \cdot \mathbf{E} \rightarrow \mathbf{j} \cdot \overleftrightarrow{\eta} \cdot \mathbf{j}$$ which is often approximated to be ~$\eta \ j^{2}$, where we have reduced the tensor to a scalar. In this form, one would call this Ohmic heating or resistive heating. I think of it this way because the conversion of $\mathbf{E}$ to a function of $\mathbf{j}$ is referred to as Ohm's law.

Drag Force
Drag forces are often written in a form similar to: $$\mathbf{F} = - b \ \mathbf{v}$$ where $b$ is a constant and $\mathbf{v}$ is the velocity of the object experiencing the drag. In a collisional medium, $b$ $\rightarrow$ $m \ \nu$, where $m$ is the mass of the object and $\nu$ is a characteristic frequency which is a collision rate in this case.

The advantage of this form, $\mathbf{F}$ = -$m \ \nu \ \mathbf{v}$, is that $\nu$ can be applied to binary collisions, Coulomb collisions, or wave-particle collisions (what I referred to as anomalous collisions before).

Relation to Resistivity
The collision frequency can be related to resistivity by: $$\eta = \frac{ m \ \nu }{ n_{e} \ e^{2} }$$ where $n_{e}$ is the electron number density and $e$ is the fundamental charge. In the ionosphere, the dominant resistive terms arise from electron-neutral and electron-ion collisions. In the solar wind, however, the Coulomb collision rates are roughly one per day near Earth (assuming $90^{\circ}$ deflections, i.e., not including small angle deflections). So in the presence of a waves/instabilities, the dominant terms arise from wave-particle collisions where the wave fields act as scattering centers.

The proper form for the wave-particle collision rate depends upon the dispersion relation for the wave. In the quasi-linear approximation for ion-acoustic waves, for instance, the anomalous collision frequency is given by: $$\nu = \omega_{pe} \frac{ \varepsilon_{o} \ \delta E^{2} }{ 2 \ n_{e} \ k_{B} \ T_{e} }$$ where $\omega_{pe}$ is the electron plasma frequency, $\delta E$ is the wave electric field, $k_{B}$ is the Boltzmann constant, and $T_{e}$ is the electron temperature.

Summary
I think the best thing to do is explicitly state the term(s) that you are referring to in your work. Meaning, write out $\mathbf{j} \cdot \mathbf{E}$ as the definition of Joule heating, for instance. If you explicitly show the term to which you refer, you will not have an issue. The confusion largely arises from implied relationships between the jargon and the actual mathematical expressions that are used incorrectly or carelessly.

I also agree with Vytenis, in that the frame of reference for which you define these terms is critical because both $\mathbf{j}$ and $\mathbf{E}$ depend upon the frame of reference. However, if you clearly define each of the terms to which you refer, this should not be an issue either.