The current calculation of Betatron acceleration of test particles follows the formula: dW/dt = μ * (∂B/∂t) where the partial derivative is found using: ∂B/∂t = dB/dt - v ⋅ ∇B. My check on the 2D XZ magnetosphere simulation shows that this term is anti-correlated with the gradient drift acceleration, which makes me thinking that this calculation is somehow dominated by the - v ⋅ ∇B term. In addition, across a shock from the upstream to the downstream, this μ * (∂B/∂t) must be positive!
I was corrected by Gemini that μ * (∂B/∂t) is not necessarily positive across the shock. dB/dt is positive since this is the change of magnetic field along the particle trajectory from the upstream to the downstream, and most often the convection term v ⋅ ∇B dominates over the total time derivative term. This indicates that ∂B/∂t = dB/dt - v ⋅ ∇B is more likely negative!
Now we added the polarization drift acceleration, which should complete all the drifts that leads to acceleration in the guiding center approximation. The remaining part is supposed to be contributed by the non-adiabatic terms.
One thing that could lead to the error in the Betatron acceleration calculation is that we only save the solution along the trajectory every a few points, so the terms like the time derivative of the total magnetic field may not be accurate. TBI
The current calculation of Betatron acceleration of test particles follows the formula: dW/dt = μ * (∂B/∂t) where the partial derivative is found using: ∂B/∂t = dB/dt - v ⋅ ∇B. My check on the 2D XZ magnetosphere simulation shows that this term is anti-correlated with the gradient drift acceleration, which makes me thinking that this calculation is somehow dominated by the - v ⋅ ∇B term.
In addition, across a shock from the upstream to the downstream, this μ * (∂B/∂t) must be positive!I was corrected by Gemini that μ * (∂B/∂t) is not necessarily positive across the shock. dB/dt is positive since this is the change of magnetic field along the particle trajectory from the upstream to the downstream, and most often the convection term v ⋅ ∇B dominates over the total time derivative term. This indicates that ∂B/∂t = dB/dt - v ⋅ ∇B is more likely negative!
Now we added the polarization drift acceleration, which should complete all the drifts that leads to acceleration in the guiding center approximation. The remaining part is supposed to be contributed by the non-adiabatic terms.
One thing that could lead to the error in the Betatron acceleration calculation is that we only save the solution along the trajectory every a few points, so the terms like the time derivative of the total magnetic field may not be accurate. TBI