Understanding Tokamak Heating: Exploring Maxwell's Equations and Magnetic Fields

[tim9000]

HI,
Some things I want to confirm, somethings I need to be reminded of:
So what I remember is that Magnetostatics - stationary charges
Electrostatics - constant or slow moving charges
And that an accelerating charge makes a magnetic field, but you need a time varying field to move charges.
So for that 'Curl of B' Maxwell Equ I gather that the partial derivative cross product of B means it's moving.

So my question is, those outter Ohmic heating coils and cores in the tokamak picture I attached, they must be oscillating in current excitation, and thus oscillating flux direction in those cores mustent they? Because that circulating flux produces the J axial current flowing inside the toroid, does that mean that the particles in the toroid are swiching back and fourth? What would you do if you didn't want the axial current to go back and fourth, but instead keeps spinning in one direction?
Is that second term 'mu*epsilon*partial derivative of E', in the maxwell equation zero? I only vaguely remember it was derived from Biot-Savart somehow with like the 'bag inside a capacitor' thing Maxwell put there to fix the equation. I don't really remember what it means.

How does the Tokamak heat at first? So plasma is conductive (hence the aformentioned J current), but how does it produce an axial current conduction before it heats up to be conductive plasma?

Finally, is it correct to say: the outter heating coils and cores, they produce an axial current inside which produces a poloidal magnetic field. And the wound coils around the toroid produce an axial magnetic field which result in a helix magnetic filed inside the tokamak?

Cheers

 

[tim9000]

Well a moving charge, not necessarily accelerating charge.

So for that 'Curl of B' Maxwell Equ I gather that the partial derivative cross product of B means it's moving.

I didn't mean B was moving, just changing.I was thinking that since it's a gas in the toroid that the particles would move anyway and heat up, so why would it need to be a conductive plasma to move?

 

[the_wolfman]

And that an accelerating charge makes a magnetic field, but you need a time varying field to move charges.

This isn't true. Recall that the force on a charged particle is given by the equation $F=q\left(\vec E + \vec V \times \vec B \right)$.
This equation implies that charged particles are accelerated by constant electric and magnetic fields.

However, electric induction does require a time varying magnetic flux to induce a potential. Tokamaks use induction to drive toroidal currents in the plasma. However time varying does necessarily not mean oscillating. In tokamaks, the change in the magnetic flux is monotonic, it does not reverse direction. During a discharge the toroidal current in a plasma only flows in one direction. It does not change direction.

How does the Tokamak heat at first? So plasma is conductive (hence the aformentioned J current), but how does it produce an axial current conduction before it heats up to be conductive plasma?

The EMF produced by induction is strong enough to cause break down of the neutral gas. The resulting plasma is relatively cold, and thus very resistive. The cold plasma is ohimically heated by the toroidal current. As it heats up the resistivity decreases, and ohmic heating is less efficient. Its at these high temperatures where some form of auxiliary heating is needed. Neural beam heating and electron cyclotron heating are two commonly used forms of auxiliary heating.

Finally, is it correct to say: the outter heating coils and cores, they produce an axial current inside which produces a poloidal magnetic field. And the wound coils around the toroid produce an axial magnetic field which result in a helix magnetic filed inside the tokamak?

This is the basic idea. Real tokamaks have multiple sets of external coils that shape both the toroidal and poloidal fields. We call the "axial" field the toroidal field.

 

[tim9000]

Recall that the force on a charged particle is given by the equation F=q(E⃗ +V⃗ ×B⃗ )F=q\left(\vec E + \vec V \times \vec B \right).
This equation implies that charged particles are accelerated by constant electric and magnetic fields.

However, electric induction does require a time varying magnetic flux to induce a potential. Tokamaks use induction to drive toroidal currents in the plasma. However time varying does necessarily not mean oscillating. In tokamaks, the change in the magnetic flux is monotonic, it does not reverse direction. During a discharge the toroidal current in a plasma only flows in one direction. It does not change direction.

Umm, sorry you're going to have to give me a phys101 remeinder; I remember the Lorenz force but to make an analogy of what I look at the outter heating coils of the tokamak to look like: if I have a wire passing through what looks like a big permanent magnet (say a wire, but which is a gas in the case of the tokamak, passing the centre of a magnet) there is not perpetual current flowing through the centre wire from the surrounding magnetic field. That is to say, if I just put a dc current to excite the Ohmic heating coils I couldn't just expect the particles in the tokamak to speed up indefinitely could I? So are you saying the Ohmic heating coils are excited with a timevarying current BUT it is only like the top half of the cycle? So it is technically DC but peaks, goes to zero, peaks repeat? And this is how the flux and particles don't reverse direction?

This is the basic idea. Real tokamaks have multiple sets of external coils that shape both the toroidal and poloidal fields. We call the "axial" field the toroidal field.

Thanks for the clarification, interesting to think of cold plasma.

 

[the_wolfman]

Umm, sorry you're going to have to give me a phys101 remeinder; I remember the Lorenz force but to make an analogy of what I look at the outter heating coils of the tokamak to look like: if I have a wire passing through what looks like a big permanent magnet (say a wire, but which is a gas in the case of the tokamak, passing the centre of a magnet) there is not perpetual current flowing through the centre wire from the surrounding magnetic field. That is to say, if I just put a dc current to excite the Ohmic heating coils I couldn't just expect the particles in the tokamak to speed up indefinitely could I? So are you saying the Ohmic heating coils are excited with a timevarying current BUT it is only like the top half of the cycle? So it is technically DC but peaks, goes to zero, peaks repeat? And this is how the flux and particles don't reverse direction?

Perhaps this will help...

In a typical tokamak discharge, the current is ramped up from zero to some nominal value. Then it is maintained at this level for as long as possible. However, you can't sustain this current forever, and eventually it will resistively decay. When it decays below some nominal value we end the discharge. There is nothing cyclic about it.

Sometimes a picture is worth a thousand words. Here is a link to a somewhat recent overview talk of the DIII-D tokamak:
https://diii-d.gat.com/diii-d_global/presentations/dpp2009/Fenstermacher.pdf
The left graph on slide 5 shows a characteristic tokamak discharge. The top pane shows plasma current labeled $I_p$ (the black trace).

There is a physical limit as to how long we can inductively maintain this current. This currently limits how long a tokamak discharge lasts. There's currently a lot of research into non-inductive current drive. Such methods would enable continuous operation of a steady state reactor.

And this is how the flux and particles don't reverse direction

For clarification: The plasma current is due to differences between electron and ion flows. The direction of the current does not necessary coincide with the direction of the particle flows.

Thanks for the clarification, interesting to think of cold plasma.

Cold is a relative term. Fusion requires plasma temperature on the order of 100 million degrees. Where "cold" plasmas usually have temperatures around 10-100 thousand degrees.

 

[tim9000]

In a typical tokamak discharge, the current is ramped up from zero to some nominal value. Then it is maintained at this level for as long as possible. However, you can't sustain this current forever, and eventually it will resistively decay.

Oh right, so what causes the resistive decay? Because as a plasma heats up it conducts better. So in practice we can't run these with circulating current or ions indefinitely yet?

Cold is a relative term. Fusion requires plasma temperature on the order of 100 million degrees. Where "cold" plasmas usually have temperatures around 10-100 thousand degrees.

When I said cold, I was thinking like the machine is off, you turn the outter magnetic cores on, then the gas inside the toroid goes from room temperature to separating and starting to circulate.

For clarification: The plasma current is due to differences between electron and ion flows. The direction of the current does not necessary coincide with the direction of the particle flows.

So the H or He will be traveling in one direction as soon as the magnetic field is applied, but the electrons won't start conducting until the gas is a plasma?I read briefly through your link, I want to have a good look at it when I have the time soon, that's going to take me ages to digest.
But just humour me here, I drew a picture of the analogy I was going to make:

What I meant was that if the toroid was a copper loop and the outter heating cores were the permanent magnet, then I wouldn't expect any current to flow unless the flux was changing. So what I was asking was, what's the difference between that analogy and the tokamak? (there's just something I am forgetting and don't get)Thanks very much, appreciate it!

 

[the_wolfman]

Oh right, so what causes the resistive decay? Because as a plasma heats up it conducts better.

While they are good conductors, plasmas are not perfect conductors. This small, but finite, resistivity causes the currents in the plasma to decay.

So the H or He will be traveling in one direction as soon as the magnetic field is applied, but the electrons won't start conducting until the gas is a plasma?

The neutral gas doesn't really respond to the applied magnetic field.

I read briefly through your link, I want to have a good look at it when I have the time soon, that's going to take me ages to digest.

I picked that talk simply because it had a nice graph of the toroidal current. I wouldn't recommend trying too hard to digest the content. Its an overview talk that is specifically designed to be quickly highlight some main results. Each slide basically summaries another talk that followed.

But just humour me here, I drew a picture of the analogy I was going to make:

A useful analogy is to consider the consider the tokamak as a transformer. The current in the primary winding comes from your power supply. The plasma in the tokamak constitutes a single turn secondary winding. The transformer is not a permanent magnet. A key point is that the flux in the core has to change to induce a current in the plasma.

 

[tim9000]

A key point is that the flux in the core has to change to induce a current in the plasma.

That is exactly my point: I asked how can the plasma current conduct in the same direction if their was changing flux and I possited:

So are you saying the Ohmic heating coils are excited with a timevarying current BUT it is only like the top half of the cycle? So it is technically DC but peaks, goes to zero, peaks repeat? And this is how the flux and particles don't reverse direction?

This is my main point of confusion, thanks a lot

 

[the_wolfman]

So the point I'm trying to emphasis is that the current that you supply to the transformer and the current induced in the plasma are not cyclical.
A current that peaks, goes to zero, then peaks again is cyclical. It repeats itself!

A cyclical current would be for a tokamak. The toroidal current produces a poloidal magnetic field. The polodial magnetic field is important for force balance and stability. If you apply a cyclical current, there will be a phase of the cycle where the inductive EMF opposes the plasma current. During this phase the plasma current will decrease causing the poloidal field to decrease. This decrease in the polodial field will causes the tokamak to lose stability!

We first learn about transforemrs in the contex of AC currents. But all a transforemr needs to induce a current in the secondary winding is a time varying primary current. There are plenty of time varying currents that are not cyclical. For example consider a linear current, an exponential current, or a hyperbolic tangent current. If you supply any of the these currents to a transformer, they will still induce a current in the secondary winding. Yet none of these wave forms are cyclical. Their derivatives are all strictly positive.

 

[tim9000]

Ahh, I see, hmm.

We first learn about transforemrs in the contex of AC currents. But all a transforemr needs to induce a current in the secondary winding is a time varying primary current. There are plenty of time varying currents that are not cyclical. For example consider a linear current, an exponential current, or a hyperbolic tangent current. If you supply any of the these currents to a transformer, they will still induce a current in the secondary winding. Yet none of these wave forms are cyclical. Their derivatives are all strictly positive.

I imagine the Amperage of the coils and Magnetic field in the outter heating coils just ramping up forever, to induce the current in the Tokamak (the secondary) if it is fed with an exponential current for instance. But surely this cannot continue for ever as the limits of the current the wire can take and the magnetic saturation of the cores would surely curtail this?

Very interesting discussion, thanks