# Plasma Current

## Main Question or Discussion Point

Hello all,

Say we ignore plasma instabilities and Lawson's criterion, what is the minimum current required (or how do I calculate it) to create fusion/plasma? For a deuterium-deuterium reaction, the potential barrier to overcome is 7.6*10^(-14) Joules and temperature T=3.6*10^9 K.

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mfb
Mentor
Current? Do you mean temperature? There is no minimum anything, fusion rate just grows gradually as conditions get better. There are tables about fusion rates as function of temperature. The dependence on pressure is easier to handle in a formula.

I mean how are current and temperature related? I know the concepts of reaction rate, Lawson's criterion, instabilities etc. Let's say, theoretically, that I need 20 kA plasma current. How do I find to what temperature this current corresponds? Perhaps convert both quantities to energy as in joules and compare them and then turn the joules to keV to find my temperature?

mfb
Mentor
I mean how are current and temperature related?
They don't have a specific relation.
Perhaps convert both quantities to energy as in joules and compare them and then turn the joules to keV to find my temperature?
That does not make sense.

Ok can you explain to me how when trying to achieve fusion, the Ohmic input heating via induced current by a transformer it is said to raise the plasma to temperatures of 4-5 keV?

mfb
Mentor
It is done that way in some fusion reactors, and differently in others. That does not imply a special relation between temperature values and current values.

In a plasma, ohmic resistance usually decreases when temperature rises - just the opposite as in most metals. Therefore, at some point the maximum current is what the power source can deliver but increasing the current will not heat the plasma much further. The exact quantities depend on a lot of things - plasma composition, dimensions, density. This point is in a reactor too low to create enough fusion for net energy gain, so additional heating methods are applied.

Yes I know that. And I am aware of other limitations as well. The only thing I don't understand is the current-plasma heating concept. The plasma works as a one-turn secondary winding for the transformer. So it is subjected to whatever current is coming from the system. But how is it actually heated. Any thoughts on the math-physics involved?

In a tokamak the plasma acts as the secondary winding of a transformer, in a stellarator it doesn't so the other methods apply. On a low level: the current creates the plasma in a tokamak: the transformer creates a strong EM field, which strips some electrons from their nuclii. Those get accelerated, collide with other atoms and molecules, strip electrons from them too who also get accelerated, etc. etc. This is just an electrical discharge, which is easy to describe in words but can be rather complicated to describe mathematically because of instabilities.

Is this what you wanted to know or did I misunderstood the question?

Yes I know that. And I am aware of other limitations as well. The only thing I don't understand is the current-plasma heating concept. The plasma works as a one-turn secondary winding for the transformer. So it is subjected to whatever current is coming from the system. But how is it actually heated. Any thoughts on the math-physics involved?
If you drive a current through a plasma it will heat up due to the resistance of the plasma. The Ohmic heating power is $\eta J^2$. This is the same as what happens in wire. If you drive current through a wire, the resistance of the wire will produce heat at the rate $P=I^2 R$.

The resistance of the plasma actually depends on the temperature. One common model of the resistivity is the Spitzer model which states the $\eta = \eta_0 T^{-3/2}$. The Spitzer model is pretty good for colder collisional plasmas, but it breaks down for hotter collisionless plasmas.

However, what you want to know it the temperature of the plasma. If you want to calculate the temperature of any object you have to perform a of thermal transport analysis. Transport calculations require multiple pieces of information. You need to know the heating sources (in your case this is Ohmic power), you also need to know the boundary conditions (what is the temperature at the edge of you plasma), you need to know all the sources of the thermal transport (in a plasma you have a collisional heat flux, a neoclassical heat flux, a turbulent heat flux, a radiative heat flux, etc) and you need to know the geometry of object your studying (in a plasma this is defined by the equilibrium magnetic field).