docs: add P3 multiple solutions example

docs/src/P3Scheme.md

@@ -284,13 +284,14 @@ which looks like this:
 
 ```@example
 include("plots/P3SlopeParameterizations.jl")
+
+nothing # hide
 ```
 ![](P3SlopeParameterizations_power_law.svg)
 
-With this choice, it appears that some values of $\log(L/N)$ gives rise to multiple solutions for $λ$, as seen in the plot below.
-
-!!! todo "TODO: Add plot of multiple solutions"
+With this choice, it appears that some values of $\log(L/N)$ gives rise to multiple solutions for $λ$, as seen in the plot below. Each vertical line shows a different solution for $λ$ for a given value of $\log(L/N)$. The right panel shows the number concentration distribution $N'(D)$ for the different solutions.
 
+![](P3SlopeParameterizations_multiple_solutions.svg)
 
 #### $μ$ as a constant
 

docs/src/plots/P3SlopeParameterizations.jl

@@ -3,6 +3,7 @@ import CloudMicrophysics as CM
 import ClimaParams as CP
 import CloudMicrophysics.Parameters as CMP
 import CloudMicrophysics.P3Scheme as P3
+import Printf
 
 FT = Float64
 
@@ -12,6 +13,7 @@ function make_slope_plot(slope_law, title)
     fig = Makie.Figure(size = (400, 300), figure_padding = 20)
 
     ax = Makie.Axis(fig[1, 1]; title, xscale = log10, xlabel = "λ", ylabel = "μ")
+    Makie.vlines!(λ_bnds, color = (:gray, 0.5))
 
     Makie.lines!(ax, exp.(logλs), P3.get_μ.(slope_law, logλs))
 
@@ -20,18 +22,74 @@ end
 
 
 # Power law parameterization
-slope_power_law = CMP.ParametersP3(FT).slope
+params = CMP.ParametersP3(FT)
+slope_power_law = params.slope
+
+log_λ_from_μ(spl::CMP.SlopePowerLaw, μ) = log((μ + spl.c) / spl.a) / spl.b
+λ_bnds = log_λ_from_μ.(slope_power_law, [0.0, 6.0]) .|> exp
 
 fig = Makie.with_theme(Makie.theme_minimal()) do
     make_slope_plot(slope_power_law, "μ as a function of λ (power law)")
 end
 Makie.save("P3SlopeParameterizations_power_law.svg", fig)
 
-# Constant parameterization
-# params = CMP.ParametersP3(FT; slope_law = :constant)
-# slope_constant = params.slope
+# Multiple solutions issue
+
+function make_multiple_solutions_plot()
+    state = P3.get_state(params; F_rim = 0.0, ρ_r = 400.0)
+    L_known = 2.39e-4
+    N_known = 1e5
+    target_log_L_div_N = log(L_known) - log(N_known)
+    shape_problem(log_λ) = P3.log_L_div_N(state, log_λ) - target_log_L_div_N
+    shape_sol = shape_problem.(logλs)
+    # Brute-force roots
+    dists = P3.get_distribution_parameters_all_solutions(state; L = L_known, N = N_known)
+
+    # Make figure
+    fig = Makie.Figure(size = (800, 300), figure_padding = 20)
+    ax = Makie.Axis(fig[1, 1];
+        title = "Valid solutions of shape solver",
+        xscale = log10, xlabel = "λ", ylabel = "shape solution", limits = ((10^3.55, 10^4.7), (-0.5, 0.5)),
+    )
+    Makie.vlines!(λ_bnds, color = (:gray, 0.5))
+    Makie.hlines!(ax, 0; color = (:gray, 0.2))
+
+    Makie.lines!(ax, exp.(logλs), shape_sol; color = :black)
+    map(dists) do dist
+        λ = exp(dist.log_λ)
+        Makie.vlines!(ax, λ; label = Printf.@sprintf("λ = %.0f m⁻¹", λ))
+    end
+    Makie.axislegend(ax; framevisible = true)
+
+    # N_ice axis
+    mm_to_m = 1e-3 # m
+    m_to_mm = 1e3  # mm
+    m_to_cm = 1e2  # cm
+    Ds = 10.0 .^ (-4:0.001:1.0) * mm_to_m  # in m
+    Ds_mm = Ds * m_to_mm # for plotting
+    ax_opts = (;
+        limits = (extrema(Ds_mm), nothing),
+        xscale = log10,
+        xlabel = "D (mm)",
+    )
+    Makie.Axis(
+        fig[1, 2];
+        ax_opts...,
+        # ylabel = L"$N_{ice}$ (1/cm³)", 
+        ylabel = "[1/cm⁴]",
+        title = "Number concentration distribution, N'(D)",
+    )
+    Makie.vlines!(state.D_th * m_to_mm, color = (:gray, 0.5))
+    for dist in dists
+        N′ = @. exp(P3.log_N′ice(dist, Ds))
+        Makie.lines!(Ds_mm, N′ / m_to_cm^4)
+        # Makie.lines!(Ds_mm, cumsum(N′.(Ds) .* Ds / m_to_cm^3))  # CUMULATIVE DISTRIBUTION
+    end
 
-# fig = Makie.with_theme(Makie.theme_minimal()) do
-#     make_slope_plot(slope_constant, "μ as a constant")
-# end
-# save("P3SlopeParameterizations_constant.svg", fig)
+    return fig
+end
+
+fig = Makie.with_theme(Makie.theme_minimal()) do
+    make_multiple_solutions_plot()
+end
+Makie.save("P3SlopeParameterizations_multiple_solutions.svg", fig)