Simple air-sea flux calculation

Simple air sea flux calculation and it's adjoint, obtained via Enzyme.

using ECCO
(f,f_ad,x)=toy_problems.Enzyme_ex1()
f_x=f(x); f_ad_x=f_ad(x)[1][1]
println("f(x)=$f_x adfx=$f_ad_x")

f(x)=-3.858024691358025e-6 adfx=3.858024691358025e-6

Bulk formulae

Air sea flux calculation derived using standard bulk formulae algorithm, and it's adjoint, obtained via Enzyme.

(f,f_ad,x)=toy_problems.Enzyme_ex2()
f_x=f(x...); f_ad_x=f_ad(x...)
println("f(x)=$(f_x) adfx=$(f_ad_x)")

f(x)=1.2244888899523058e-9 adfx=((1.2244888899523058e-9, 8.970050297237849e-11),)

(f,f_ad,x,y)=toy_problems.Enzyme_ex3()
f_x=f(x,y); f_ad_x=f_ad(x,y)
println("f(x)=$(f_x) adfx=$(f_ad_x)")

f(x)=0.00915548218468587 adfx=[0.0, 0.0, 0.01831096436937174, 0.0]

(f,f_ad,x,y)=toy_problems.Enzyme_ex4()
f_x=f(x,y); f_ad_x=f_ad(x,y)
println("f(x)=$(f_x) adfx=$(f_ad_x)")

f(x)=-3.0606099804357885 adfx=[0.0, 917.7850566361462, -6.121219960871577, -0.44841281435892]

(x,adx)=toy_problems.ForwardDiff_ex1()
println("x=$(x) adfx=$(adx)")

x=[300.0, 0.001, 1.0, 10.0] adfx=[0.0, 458.8925283180731, -3.0606099804357885, -0.22420640717946003]

(x,adx)=Zygote_examples.Zygote_ex1()

([300.0, 0.001, 1.0, 10.0], [0.0, 458.8925283180731, -3.0606099804357885, -0.22420640717946])

Optimization

using ECCO
(f,x0,x1,result)=toy_problems.optim_ex1()

(ECCO.toy_problems.var"#f#6"(), [0.0, 0.0], [0.9999634355313174, 0.9999315506115275],  * Status: success

 * Candidate solution
    Final objective value:     3.525527e-09

 * Found with
    Algorithm:     Nelder-Mead

 * Convergence measures
    √(Σ(yᵢ-ȳ)²)/n ≤ 1.0e-08

 * Work counters
    Seconds run:   0  (vs limit Inf)
    Iterations:    60
    f(x) calls:    117
)

h,h!,x0,x1,result=toy_problems.optim_ex2()

(ECCO.toy_problems.var"#f#7"(), ECCO.toy_problems.var"#g!#8"(), [0.0, 0.0], [0.999999999999928, 0.9999999999998559],  * Status: success

 * Candidate solution
    Final objective value:     5.191703e-27

 * Found with
    Algorithm:     L-BFGS

 * Convergence measures
    |x - x'|               = 4.58e-11 ≰ 0.0e+00
    |x - x'|/|x'|          = 4.58e-11 ≰ 0.0e+00
    |f(x) - f(x')|         = 4.41e-19 ≰ 0.0e+00
    |f(x) - f(x')|/|f(x')| = 8.50e+07 ≰ 0.0e+00
    |g(x)|                 = 1.44e-13 ≤ 1.0e-08

 * Work counters
    Seconds run:   0  (vs limit Inf)
    Iterations:    24
    f(x) calls:    67
    ∇f(x) calls:   67
)

h,h!,x0,x1,result=toy_problems.optim_ex3()

(ECCO.toy_problems.var"#h#9"(), ECCO.toy_problems.var"#h!#10"{ECCO.toy_problems.var"#h#9"}(ECCO.toy_problems.var"#h#9"()), [0.0, 0.0], [0.999999999999928, 0.9999999999998559],  * Status: success

 * Candidate solution
    Final objective value:     5.191703e-27

 * Found with
    Algorithm:     L-BFGS

 * Convergence measures
    |x - x'|               = 4.58e-11 ≰ 0.0e+00
    |x - x'|/|x'|          = 4.58e-11 ≰ 0.0e+00
    |f(x) - f(x')|         = 4.41e-19 ≰ 0.0e+00
    |f(x) - f(x')|/|f(x')| = 8.50e+07 ≰ 0.0e+00
    |g(x)|                 = 1.44e-13 ≤ 1.0e-08

 * Work counters
    Seconds run:   0  (vs limit Inf)
    Iterations:    24
    f(x) calls:    67
    ∇f(x) calls:   67
)

Lorenz

See https://en.wikipedia.org/wiki/Lorenz96model

using ECCO, CairoMakie
xyz=Lorenz_models.L96()
lines(xyz[1,:],xyz[2,:],xyz[end,:])

x,y,z=Lorenz_models.L63()
lines(x,y,z)

Budyko-Sellers Energy Balance Model

See this tutorial for detailed explanations.

using ECCO, CairoMakie

(; Q, y) = Budyko_Sellers_models.params
Tsol,Tini,dTdt_ini,incr_t,incr=Budyko_Sellers_models.dTdt_demo(Q)

fig=Figure()
Axis(fig[1,1]); lines!(y,dTdt_ini)
Axis(fig[2,1]); lines!(y,Tini); lines!(y,Tsol,color=:red)
Axis(fig[3,1]); lines!(incr_t,incr)
fig

Parameter Choices

From B. Rose notebook :

R = 10^7 J/m2/K

From Walsh and Rackauckas :

• Figure 3. Equilibrium solutions (7) with albedo function (9) for five η-values. Note T∗ ηi (ηi) = Tc only for i = 2,5.
  • Parameters: Q = 343, A = 202, B = 1.9, C = 3.04, αw = 0.32, αs = 0.62, Tc =−10.
• Figure 9. Equilibrium solutions of (2) with albedo function (34). Solid: η= 0.1. Dashed: η= 0.25. Dash-Dot: η= 0.4.
  • Parameters: Q = 321,A = 167,B = 1.5,C = 2.25,M = 50,αw = 0.32,αi = 0.46,αs = 0.72,ρ= 0.35.

Earlier Implementations

• https://brian-rose.github.io/ClimateLaboratoryBook/courseware/one-dim-ebm.html
• https://github.com/ECCO-Summer-School/ESS25-Team_FLOW
• https://www.cise.ufl.edu/~luke.morris/2_4_2025/build/bsh/budyko_sellers_halfar/

Related References

• discrete and continuous – on the Budyko-Bellers energy balance climate model with ice line coupling – James Walsh, Christopher Rackauckas – doi:10.3934/dcdsb.2015.20.2187
• Theory of Energy-Balance Climate Models – Gerald R. North – 1975 – DOI: <https://doi.org/10.1175/1520-0469(1975)032<2033:TOEBCM>2.0.CO;2>
• Predictability in a Solvable Stochastic Climate Model – Gerald R. North and Robert F. Cahalan – DOI: https://doi.org/10.1175/1520-0469(1981)038%3C0504:PIASSC%3E2.0.CO;2

Glacier

Simple, 1D mountain glacier model inspired from the book Fundamentals of Glacier Dynamics, by CJ van der Veen, and which was translated to Julia by S Gaikwad.

See https://sicopolis.readthedocs.io/en/latest/AD/tutorial_tapenade.html#mountain-glacier-model

using ECCO
V=glacier_model.forward_problem(0.002)

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