In plasma, such as in the solar system’s plasma, quantities that change very slowly in time and space, i.e., remain nearly constant within specific limits or over time, are called adiabatic invariants. A good example is the magnetic moment $\mu=W_\perp/B$, which we have encountered while discussing magnetic drift, particularly gradient drift.

In a plasma, for every type of motion of a particle, there is an adiabatic invariant. For instance, the invariant related to the particle’s gyration is the magnetic moment $\mu$. The longitudinal motion along the magnetic field lines corresponds to the longitudinal invariant $J$, and the motion involving all perpendicular drifts to the magnetic field corresponds to a third invariant: the drift invariant $\Phi$.

When these three types of motion are periodic, and the angular frequency of system changes is much lower than the average oscillation frequency of various particles, the Hamiltonian action integral

$$ J = \oint p_i dq_i $$

remains unchanged over time, and this action is referred to as an adiabatic invariant. This discussion encompasses all three types of invariants.

The magnetic moment $\mu$ can be proven invariant or constant using the energy conservation principle. The sum of the particle’s kinetic energy parallel and perpendicular to the magnetic field can never change, meaning

$$ \frac{dW_\parallel}{dt} + \frac{d W_\perp}{dt} = 0$$

However, deriving the relationship of these energies with the magnetic moment shows that

$$ \frac{dW_\parallel}{dt} + \frac{d W_\perp}{dt} = B\frac{d\mu}{dt} = 0$$

i.e., the magnetic moment remains constant. When a particle’s magnetic field changes (spatially), the frequency and radius of its gyration adjust so that the magnetic moment remains unchanged.

Spatial variations in the electric field cause temporal changes in the magnetic field, as $\partial \mathbf{B}/dt=-\nabla\times E$. The rate of temporal change of the magnetic field is proportional to the curl of the electric field and works in the opposite direction. However, even these spatial variations in the electric field or temporal changes in the magnetic field do not affect the magnetic moment.

Similarly, temporal variations in the electric field induce second-order temporal changes in the magnetic field, which are negligible.

If the magnetic moment is invariant, the magnetic flux perpendicular to the gyro orbit must also be invariant. The magnetic flux $\Phi_\mu = B\pi r_g^2$, where the gyro radius $r_g=mv_\perp/(qB)$, gives:

$$ \Phi_\mu = \pi B \left(\frac{m v_\perp}{qB}\right)^2 = \pi B \frac{mv_\perp^2}{2B} \frac{2m}{q^2B} = \frac{2\pi m}{q^2} \mu $$

which will also be invariant like the magnetic moment. As a particle moves toward a stronger magnetic field, its gyro orbit shrinks to keep the magnetic flux enclosed by the orbit unchanged.

For a particle moving through an inhomogeneous magnetic field, the magnetic moment is expressed as:

$$ \mu = \frac{mv^2\sin^2\alpha}{2B} $$

where $v_\perp=v\sin\alpha$ and the pitch angle $\alpha = \tan^{-1}(v_\perp/v_\parallel)$. In this equation, only the magnetic moment and total kinetic energy remain unchanged, so the pitch angle changes with the magnetic field:

$$ \sin^2 \alpha \propto B $$

Thus, knowing a particle’s pitch angle at one point allows the calculation of its pitch angle at other points along the magnetic field.

The Earth’s magnetic field converges, meaning its field lines emerge from one pole and converge at the other. As a particle approaches the poles, the magnetic field becomes stronger, and the pitch angle increases. When $\alpha=90^\circ$, the particle reverses direction since all energy will then be perpendicular, with no parallel energy. The energy that reverses the particle’s motion comes from the gradient force $F_\nabla = -\mu \nabla B$, also known as the mirror force.

In the Earth’s dipolar magnetic field, this reflection process continues between poles, trapping particles gyrating around the magnetic field lines. The pitch angle of these trapped particles is determined relative to the magnetic field at the mirror point $B_m$ as follows:

$$ \sin\alpha = \sqrt{\frac{B}{B_m}} $$

This process keeps particles bouncing along field lines.

Even for particles drifting from one magnetic field line to another, the magnetic moment plays a crucial role. If the magnetic moment is invariant, i.e., constant:

$$ W_\perp \propto B $$

The perpendicular kinetic energy increases with the magnetic field. However, in this case, no reflection occurs. As particles drift toward stronger fields, their perpendicular kinetic energy increases, i.e., the temperature rises. This phenomenon is termed adiabatic heating and is a type of betatron acceleration. Since only perpendicular energy increases, anisotropy develops.

The longitudinal invariant arises in dipole fields with mirror symmetry, where field lines converge at both poles. In such fields, particles bounce between poles at a specific bounce frequency $\omega_b$. The longitudinal invariant is given by:

$$ J = \oint m v_\parallel ds $$

where $v_\parallel$ is the velocity parallel to the field, and the integration is performed over the particle’s trajectory $ds$ for a complete oscillation. If the frequency of electromagnetic changes is much lower than the bounce frequency, the longitudinal invariant remains constant.

If the distance between two mirror points near the poles is $l$, and the average parallel velocity of the particle between them is $\langle v_\parallel \rangle$, then:

$$J=2ml \langle v_\parallel \rangle$$

as the particle traverses this distance twice in one oscillation. If the particle drifts to another field line, this distance changes, and the parallel kinetic energy changes accordingly. Squaring the above equation shows that:

$$ \langle W_\parallel \rangle \propto l^{-2} $$

i.e., as the distance decreases, the parallel kinetic energy increases. This forms the basis of Fermi acceleration. Combining it with the perpendicular kinetic energy seen earlier, anisotropy can be defined as:

$$ A_W = \frac{\langle W_\perp \rangle}{\langle W_\parallel \rangle} $$

And comparing it with the relationship of perpendicular energy in magnetic mirrors:

$$ A_W \propto Bl^2 $$

Thus, anisotropy increases only if the square of the field line’s length decreases less than the increase in the magnetic field.

The previously mentioned magnetic flux is the drift invariant, particularly for drift motion. To explain: as noted earlier, particles trapped in Earth’s magnetic field have three types of motion: gyration, bounce, and drift. Drift occurs perpendicular to the magnetic field, from one field line to another. If a particle drifts around the Earth’s magnetic field axis in a complete loop, its path forms a drift shell. The magnetic flux enclosed by this shell remains unchanged, i.e., it is invariant. The drift invariant is given by:

$$ \Phi = \oint v_d r d\psi $$

where $v_d$ is the sum of all perpendicular drift velocities, $r$ is the radius of the drift shell, and $\psi$ is the azimuthal angle. Integration is performed over 0 to 360 degrees around the drift shell. The flux remains invariant as long as the electromagnetic field’s frequency is much lower than the drift frequency.

From the earlier calculation of magnetic flux for the gyro orbit:

$$ \Phi = \frac{2\pi m}{q^2} M $$

where $M$ is the magnetic moment with respect to the axisymmetric field, i.e., the magnetic moment for the drift shell.

These three invariants are valid only within specific frequency limits. The magnetic moment remains constant only if the electromagnetic frequency $\omega$, i.e., the rate of change of the electric and magnetic fields, is lower than the gyro frequency. If $\omega>\omega_g$, the magnetic moment is no longer invariant. In such cases, the guiding center concept breaks down, and the motion cannot be described simply by gyration.

The longitudinal invariant holds true only if the electromagnetic frequency lies between the gyro frequency and bounce frequency. That is, if $\omega_g > \omega>\omega_b$, the invariant no longer holds, and the particle’s motion between mirror points cannot be described as simple bouncing.

Similarly, the drift invariant becomes invalid if the electromagnetic frequency lies between the gyro frequency and bounce frequency but exceeds the drift frequency, i.e., $(\omega_g,\omega_b)>\omega>\omega_d$. In this case, the particle undergoes gyration and bounce but does not drift along the drift shell; instead, it diffuses randomly.