Mathematical and Information Technologies, MIT-2016 Mathematical modeling

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Sydney Hall
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3 ρ u,t = p,x q,y, ρ v,t = q,x p,y, ω,t = 2 q + µ x,x + µ y,y, φ,t = ω, p,t = k u,x + v,y + β T,t, q,t = α v,x u,y 2 α ω + q/η, µ x,t = γ ω,x, µ y,t = γ ω,y, c T,t = 11 T,x + 12 T,y,x + 12 T,x + 22 T,y,y β T u,x + v,y + 2 q 2 /η. u v ω φ p q µ x µ y T ρ k α η c β = 1 cos 2 φ + 2 sin 2 φ 12 = 1 2 sin φ cos φ 22 = 1 sin 2 φ + 2 cos 2 φ 1 2 t x y x y ρ u,y v,x,t = q, µ x,t µ y,t t q ω q,tt + 2 α η q,t + 2 α ω,t = α ρ q, ω,tt 2 q,t = γ ω. 475

4 q t=0 = q 0, q t=0,t = α v,x 0 u 0,y 2 α ω 0 + q0 = 2 α ω 0 + q0, η η ω t=0 = ω 0, ω t=0,t = 1 2 q 0 + µ 0 x,x + µ 0 2 q 0 y,y =, u 0 v 0 ω 0 q 0 µ 0 x µ 0 y q ω,t t ω,t = 1 α 2 α ρ q q,tt 2 α η q,t, ω,ttt = 2 q,tt + γ ω,t, q,tttt + 2 α η q,ttt + 4 α α q,tt ρ + γ q,tt 2 α γ η q,t = α ρ q t=0 = q 0, q,t t=0 = α v 0,x u 0,y 2 α ω 0 + q0 = 2 α η γ 2 q. ω 0 + q0 η q t=0,tt = α ρ q0 2 α 2 q 0 + µ 0 x,x + µ 0 2 α [ α α y,y η q0,t = 4 α η ω0 + η 2 1 q 0], q,ttt t=0 = α ρ q0,t 2 α = 8 α 2[ 1 α η 2 ω η 2 q 0,t + γ ω 0 2 α η q0,tt = η α η 2 q 0]., q ω q ω q,x ω,x q,y ω,y x y t q n+1 2 q n 1, 2 + q n 1 t 2 + α η = α ρ q n 1+1, 2 2 q n + q n 1 1, 2 x 2 q n+1 q n 1 t + α ωn+1 ω n 1 t = + qn 2 1, 2+1 qn + q n 1, 2 1 y 2, 476

5 = γ ω n+1 2 ω n 1, 2 + ω n 1 t 2 1 ω n 1+1, 2 2 ω n + ω n 1 1, 2 x 2 1 = 2, N = 2, N 2 1 ω n+1 ω n+1 = 2 ω n 1, 2 ω n 1 + t q n+1 q n 1 t = + ωn +1 2 ωn + ω n 1 y 2, q n+1 q n γ t2 ω n 1+1, 2 2 ω n 1, 2 + ω n 1 1, 2 x 2 + ωn 2 1, 2+1 ωn + ω n 1, 2 1 y 2. q n+1 + α ρ α γ t α + α η t + 1 t 2 q n+1 = 2 t 2 qn + α + + α η t 1 t 2 q n α ω n 1 t 1, 2 ω n 1, 2 + q n 1+1, 2 2 q n 1, 2 + q n 1 1, 2 + qn 2 1, 2+1 qn + q n 1, 2 1 y 2 x 2 ω n 1+1, 2 2 ω n 1, 2 + ω n 1 1, 2 x 2 + ωn +1 2 ωn + ω n 1 y 2. η q n = λ n ˆq e i1α1+2α2, ω n = λ n ˆω e i1α1+2α2. λ n e i1α1+2α2 λ 2 + 1/λ t 2 ˆq + α λ 1/λ ˆω = α e iα e iα1 t ρ x 2 + eiα2 2 + e iα2 y 2 ˆq. 477

6 λ 2 2 λ + 1 t α sin 2 ρ λ α 1 /2 x 2 + sin2 α 2 /2 y 2 ˆq + α λ2 1 ˆω = 0. t λ 2 2 λ + 1 t 2 Mathematical and Information Technologies, MIT-2016 Mathematical modeling + 4 γ λ sin 2 α 1 /2 x 2 + sin2 α 2 /2 y 2 ˆω 1 λ 2 1 t ˆq = 0. ˆq ˆω λ 1 2 t α λ 1λ + 1 λ A α ρ t 1 λ 2 1 λ t t γ = 0, λ A A = sin2 α 1 /2 x 2 + sin2 α 2 /2 y 2 a = α ρ A t2, b = γ A t2, c = α t2, λ λ λ 1 2 a + b + 16 λ 2 a b + λ c = 0, 1 + cλ λ λ 1 2 a + b λ 2 a b = 0. a b c b = 0 1+cλ λ a 1 = 0 λ 2 +2 λ a +1 = c λ 1 = λ 2, λ 1 = λ 2 = 1, a 2 1 0, 1 + c a 1 a 1, 2 2 0, a c 1 + c α 1, α 2 α 1 ρ t2 x y 2 a = 0 1+cλ λ b 1 = 0 b 1 γ 1 t2 x y 2 478

7 a + b = cλ λ 2 a b = 0 z = λ 2 z 2 2 z 1 8 a b + 1 = c z 1 = z 2 = 1, 1 8 a b 2 a b 1 0, , 1 + c 1 + c 4 a b 1 + c, 4 a b a + b 2 + c, 0 a b 2 + c. a + b = 1 α ρ + γ sin t 2 2 α 1 /2 x 2 + sin2 α 2 /2 1. y 2 λ = 1 α 1 α 2 α ρ + γ t 2 1 x y 2 479

8 x y 480

9 30 25 tcpu/tgpu N 481

10 N N N t CP U /t GP U q = ˆq e ift ky q = ˆq e ift ky ω = ˆω e ift ky f i α f η + α k2 ˆq + 2 i α f ˆω = 0, ρ 2 i f γ k 2 ˆq + f 2ˆω = 0. k ± k ± = ρ f 2 α γ d ± d 2 4 α γ f ρ 2 2 i α f η 4 α α, d = ρ +γ f 2 i α γ η. f k ± = k 1 ± + i k± 2 c ± f f = ν = k ± 2 π λ ± = 1 k ± ν k + k ν = 1 α π ρ = 1022 / 3 = / α = γ = 10 µ η = 10 ν = µ ˆq = e k2yˆq 1 cosft k 1 y ˆq 2 sinft k 1 y, ˆω = e k2yˆω 1 cosft k 1 y ˆω 2 sinft k 1 y. ω y k + k k k 482

11 c + c ν ν k + k λ + µm λ µm ν ν k + k ω ω y µm y µm k + k 483

Mathematical and Information Technologies, MIT-2016 Mathematical modeling Π 8 q = q y y c l q = 0 y y c > l ω,x = 0 y c l y c = 20 µ l = 10 µ 100 µ 40 µ 2560 1024

12 Mathematical and Information Technologies, MIT-2016 Mathematical modeling Π 8 q = q y y c l q = 0 y y c > l ω,x = 0 y c l y c = 20 µ l = 10 µ 100 µ 40 µ ρ = 1022 / 3 = / α = γ = 1 η = 100 Π q = q δx δt q = q x x c x x c = 7 µ t t q = 0 ω,y = 0 q = 0 ω = 0 10 µ 4 µ = / γ = 10 µ 484

13 q = q sin2 π ν t x x c l q = 0 x x c > l x c = 5 µ l = 2.5 µ ω,y = 0 ν ν =

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Ocean Modeling - EAS Chapter 2. We model the ocean on a rotating planet

Ocean Modeling - EAS Chapter 2. We model the ocean on a rotating planet Ocean Modeling - EAS 8803 We model the ocean on a rotating planet Rotation effects are considered through the Coriolis and Centrifugal Force The Coriolis Force arises because our reference frame (the Earth)