Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere - Simulation and Modeling Methodologies, Technologies and Application - page 62

∂ϕ = c 1 cos 2 t ∂ϕ − sin ξ sin ϕ ,

∂T

cos ϕ cos ξ ∂ξ

2

∂ξ
∂ϕ

∂ 2 T
∂ϕ 2 = c 1 cos 2 t

− 2sin ϕ cos ξ ∂ξ

∂ϕ − sin ξ cos ϕ

1+

+2 cos ξ cos ϕ

,

cos 3 κ 1 ϕ + θ 2 κ 2 sin γt sin κ 2 ϕ

sin κ 1 ϕ

−ωθ 1 κ 1

cos 3 κ 2 ϕ

1

κ 2 =0 . 1

0

−1

− 1 .5 −1 −0.5 0 0.5 1 1.5

1

κ 2 =0 . 5

0

−1

− 1 .5 −1 −0.5 0 0.5 1 1. 5

1

κ 2 =0 . 9

0

−1

−1.5 −1 −0.5 0 0.5 1 1.5

ϕ

Fig. 7. Experiment 3: graphs of the function sin( θ 2 tan κ 2 ϕ ) at θ 2 =20 , κ 2 =0 . 1 , 0 . 5 , 0 . 9

∂ϕ = −ω θ 1 κ 1

∂ξ

cos 2 κ 1 ϕ +sin γt θ 2 κ 2 cos 2 κ 2 ϕ .

Because ξ ∼ θ 2 tan κ 2 ϕ while the time grows, the term sin ξ endeavours to simulate
large gradients at the high latitudes. Indeed, while at nearly zero κ 2 's this function is
rather smooth, it becomes a saw-tooth wave as κ 2 → 1 (Fig. 7).
This experiment was performed on a series of grids 6 ◦ × 6 ◦ , 4 ◦ × 4 ◦ and 2 ◦ × 2 ◦ ,
with c 1 =10 , c 2 = 100 , ω =5 , ϑ 1 =1 , κ 1 =0 . 5 , ϑ 2 =20 , κ 2 =0 . 5 , γ =2 ,
μ =1 · 10 − 7 . At each time moment t n the numerical solution was compared with the
analytical one in the relative error

||T num − T exact || L 2 S (1)

Δλ,Δϕ

δ n = . (35)

||T exact || L 2 S (1)

Δλ,Δϕ

As it follows from (32)-(34), the forcing produces the spiral structure of the solution
near the poles, which results in rather high solution's gradients. In addition, the direc-
tion of the spiral is getting changed in time due to the term sin γt . This is exactly what
is reproduced by the numerical solution on the grid 6 ◦ × 6 ◦ shown in Fig. 8. Yet, one
can observe the cyclicity of the solution with the period 2 γ due to the sinusoid sin ξ 's
behaviour (cf., e.g., t =0 and t =3 ; t =0 . 5 and t =3 . 5 ; etc.), consistent with the term