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Force on the plasma The force on an individual particle due to the electro-magnetic field (s is species index) Assume a small volume such that Then the force per unit of volume is Force on the plasma QFor the electric field Define an average velocity Then for the magnetic fieldRRForce on the plasma]Averaged over all particles Now sum over all species The total force density therefore is ^^ Force on the plasma{This force contains only the electro-magnetic part. 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Force on the plasma The force on an individual particle due to the electro-magnetic field (s is species index) Assume a small volume such that Then the force per unit of volume is Force on the plasma QFor the electric field Define an average velocity Then for the magnetic fieldRRForce on the plasma]Averaged over all particles Now sum over all species The total force density therefore is ^^ Force on the plasma{This force contains only the electro-magnetic part. 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Break-even $The break-even condition is defined as the state in which the total fusion power is equal to the heating power Note that this does not imply that all the heating power is generated by the fusion reactions"Ignition condition$Ignition is defined as the state in which the energy produced by the fusion reactions is sufficient to heat the plasma. Only the He atoms are confined (neutrons escape the magnetic field) and therefore only 20% of the total fusion power is available for plasma heating #!n-T-tau is a measure of progress ""$Over the years the n-T-tau product shows an exponential increase Current experiments are close to break-even The next step ITER is expected to operate well above break-even but still somewhat below ignitionPForce on the plasma The force on an individual particle due to the electro-magnetic field (s is species index) Assume a small volume such that Then the force per unit of volume is Force on the plasma QFor the electric field Define an average velocity Then for the magnetic fieldRRForce on the plasma]Averaged over all particles Now sum over all species The total force density therefore is ^^ Force on the plasma{This force contains only the electro-magnetic part. For a fluid with a finite temperature one has to add the pressure force|| Reformulating the Lorentz forceBUsing The force can be written as Then using the vector identityCC Force on the plasmaOOne obtains Important parameter (also efficiency parameter) the plasma-betaPP$Writing the force in tensor notation%%$8It is convenient to write the force in a tensor notation99Tensor notation$QThe tension force Can be simplified using the divergence free magnetic field RR Tensor notation$TNow write also the pressure force as a tensor The force can then be written asUU Virial theorem tAssume an equilibrium exists, then the force must be zero Build a scalar quantity, to investigate the implication uu Virial theorem  >Using the relation One can rewrite the integral using Gauss s theorem Consider the plasma in a finite volume and extent the integration limits to infinity Virial theorem$ Z The magnetic field in the far field limit scales as So for a sufficient large radius [[ Virial theorem$ ZFor a sufficient large volume We then have Contradiction: NO equilibrium can exist"[3(/       $%&  0 `X (  x  c $ =K   x  c $ V0   X  0A%?c G X  0A&? 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Break-even $The break-even condition is defined as the state in which the total fusion power is equal to the heating power Note that this does not imply that all the heating power is generated by the fusion reactions"Ignition condition$Ignition is defined as the state in which the energy produced by the fusion reactions is sufficient to heat the plasma. Only the He atoms are confined (neutrons escape the magnetic field) and therefore only 20% of the total fusion power is available for plasma heating #!n-T-tau is a measure of progress ""$Over the years the n-T-tau product shows an exponential increase Current experiments are close to break-even The next step ITER is expected to operate well above break-even but still somewhat below ignitionPForce on the plasma The force on an individual particle due to the electro-magnetic field (s is species index) Assume a small volume such that Then the force per unit of volume is Force on the plasma QFor the electric field Define an average velocity Then for the magnetic fieldRRForce on the plasma]Averaged over all particles Now sum over all species The total force density therefore is ^^ Force on the plasma{This force contains only the electro-magnetic part. 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Break-even $The break-even condition is defined as the state in which the total fusion power is equal to the heating power Note that this does not imply that all the heating power is generated by the fusion reactions"Ignition condition$Ignition is defined as the state in which the energy produced by the fusion reactions is sufficient to heat the plasma. Only the He atoms are confined (neutrons escape the magnetic fie  ld) and therefore only 20% of the total fusion power is available for plasma heating #!n-T-tau is a measure of progress ""$Over the years the n-T-tau product shows an exponential increase Current experiments are close to break-even The next step ITER is expected to operate well above break-even but still somewhat below ignitionPForce on the plasma The force on an individual particle due to the electro-magnetic field (s is species index) Assume a small volume such that Then the force per unit of volume is Force on the plasma QFor the electric field Define an average velocity Then for the magnetic fieldRRForce on the plasma]Averaged over all particles Now sum over all species The total force density therefore is ^^ Force on the plasma{This force contains only the electro-magnetic part. 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