On the Green function of an orthotropic clamped plate in a half-plane

First, we calculate, in a heuristic manner, the Green function of an orthotropic plate in a half-plane which is clamped along the boundary. We then justify the solution and generalize our approach to operators of the form $$(Q(\partial ')-a^2\partial _n^2)(Q(\partial ')-b^2\partial _n^2)$$ ( Q ( ∂ ′ ) - a 2 ∂ n 2 ) ( Q ( ∂ ′ ) - b 2 ∂ n 2 ) (where $$\partial '=(\partial _1,\dots ,\partial _{n-1})$$ ∂ ′ = ( ∂ 1 , ⋯ , ∂ n - 1 ) and $$a>0,b>0,a\ne b)$$ a > 0 , b > 0 , a ≠ b ) with respect to Dirichlet boundary conditions at $$x_n=0.$$ x n = 0 . The Green function $$G_\xi $$ G ξ is represented by a linear combination of fundamental solutions $$E^c$$ E c of $$Q(\partial ')(Q(\partial ')-c^2\partial _n^2),$$ Q ( ∂ ′ ) ( Q ( ∂ ′ ) - c 2 ∂ n 2 ) , $$c\in \{a,b\},$$ c ∈ { a , b } , that are shifted to the source point $$\xi ,$$ ξ , to the mirror point $$-\xi ,$$ - ξ , and to the two additional points $$-\frac{a}{b}\xi $$ - a b ξ and $$-\frac{b}{a}\xi ,$$ - b a ξ , respectively.

L-shell and energy dependence of magnetic mirror point of charged particles trapped in Earth’s magnetosphere

In the Earth’s inner magnetosphere, there exist regions like plasmasphere, ring current, and radiation belts, where the population of charged particles trapped along the magnetic field lines is more. These particles keep performing gyration, bounce and drift motions until they enter the loss cone and get precipitated to the neutral atmosphere. Theoretically, the mirror point latitude of a particle performing bounce motion is decided only by its equatorial pitch angle. This theoretical manifestation is based on the conservation of the first adiabatic invariant, which assumes that the magnetic field varies slowly relative to the gyro-period and gyro-radius. However, the effects of gyro-motion cannot be neglected when gyro-period and gyro-radius are large. In such a scenario, the theoretically estimated mirror point latitudes of electrons are likely to be in agreement with the actual trajectories due to their small gyro-radius. Nevertheless, for protons and other heavier charged particles like oxygen, the gyro-radius is relatively large, and the actual latitude of the mirror point may not be the same as estimated from the theory. In this context, we have carried out test particle simulations and found that the L-shell, energy, and gyro-phase of the particles do affect their mirror points. Our simulations demonstrate that the existing theoretical expression sometimes overestimates or underestimates the magnetic mirror point latitude depending on the value of L-shell, energy and gyro-phase due to underlying guiding centre approximation. For heavier particles like proton and oxygen, the location of the mirror point obtained from the simulation deviates considerably (∼ 10°–16°) from their theoretical values when energy and L-shell of the particle are higher. Furthermore, the simulations show that the particles with lower equatorial pitch angles have their mirror points inside the high or mid-latitude ionosphere.