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zb;Q=eAt5$v8xX%#P1}73a8D*2K^2*qnekn~J}rQx$I!e7wG|Dz3csM=uZ6U_YAidV3!Qz zi1xd_(!U{flMlVJ+1q)u-8p$J(9alU(e2paB*H8JuLZa<0d;F3lO`0N=>Z^)UgF0c zD!zU*gpywlwo^<0`a)5B-72bc2G}GGpIw?ACXFhiiDI|VbwiPcdp#~Y)40j#o~1DGSEBHQG<-|Bp}$mYN{h90@Y4o0TsIY3EUnOm$)ha z;5nwiVR(<6-SDh1`Ta?w3p%;`jkKwZ1hH5{l0DsSO1m~biPMM^y)oDq02km@Hsu;K zB-qRm+3a6jF-1K2uT;bTKYsOd$2ImV9wW{~7tj^UURJu4ebMmt{{ymL-u(ao literal 0 HcmV?d00001 diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 002da03c1..17843dad0 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -1,11 +1,12 @@ # Analysing a simulation -While you can analyze a SpeedyWeather simulation through its [NetCDF output](@ref), +While you can analyze a SpeedyWeather simulation through its [NetCDF output](@ref) +(i.e., _offline_ analysis), as most users will be used to with other models, you can also reuse a lot of functionality from SpeedyWeather interactively for analysis. This makes SpeedyWeather beyond being a model for the atmospheric general circulation also a library with many functions for the analysis of simulations. -Often this also avoids the two language problem that you will face if you +Often this also avoids the two-language problem that you will face if you run a simulation with a model in one language but then do the data analysis in another, treating the model as a blackbox although it likely has many of the functions you will need for analysis already defined. @@ -18,13 +19,14 @@ and analysis are done interactively with the same library: SpeedyWeather.jl. Now you could run a SpeedyWeather simulation, and analyse the [NetCDF output](@ref) but that comes with several issues related to accuracy -- If you use a reduced grid for the simulation, then the output will (by default) be interpolated on a full grid. This interpolation comes introduces an error. +- If you use a reduced grid for the simulation, then the output will (by default) be interpolated on a full grid. This interpolation introduces an error. - Computing integrals over gridded data on the sphere by weighting every grid point according to its area is not the most accurate numerical integration. - Computing gradients over gridded data comes with similar issues. While our [RingGrids](@ref) are always equidistant in longitude, they are not necessarily in latitude. The first point you can avoid by running a simulation on one of the full grids that are implemented, see [SpectralGrid](@ref). But that also impacts the simulation and for various reasons we don't run on full grids by default. + The second point you can address by defining a more advanced numerical integration scheme, but that likely requires you to depend on external libraries and then, well, you could also just depend on SpeedyWeather.jl directly, because we @@ -35,7 +37,7 @@ spectral transforms to reduce the discretization error. The following contains a (hopefully growing) list of examples of how a simulation can be analysed interactively. We call this _online_ analysis because you are directly using the functionality from -the model as if it was a library. +the model as if it was a library. For simplicity, we use [shallow-water model](@ref shallow_water_model) to demonstrate this. ## Mass conservation @@ -47,13 +49,13 @@ The total volume is defined as the integral of the dynamic layer thickness over the surface ``A`` of the sphere ```math -\iint h dA = \iint \eta dA + \iint H dA - \iint H_b dA +M = \iint h dA = \iint \left(\eta + H - H_b\right) dA ``` to check for conservation we want to assess that ```math -\frac{\partial}{\partial t} \iint h dA = 0 +\frac{\partial M}{\partial t} = 0 ``` And because ``V = \iint H dA - \iint H_b dA``, the total volume at rest, @@ -112,14 +114,16 @@ Calculating the divergence in spherical harmonics always sets the ``l=m=0`` mode (the gradient of a periodic variable has zero mean) so the time integration here is always with an exactly zero tendency. -## Energy conservation +## Energy -The total energy in the shallow water equation is the sum of the kinetic energy -``\frac{1}{2}(u^2 + v^2)`` and the potential energy ``gh`` integrated over the -total volume (times ``h`` for the vertical then integrated over the sphere``\iint dA``). +The total energy in the shallow water equation is the sum of the kinetic energy density +``\frac{1}{2}(u^2 + v^2)`` and the potential energy density ``gz`` integrated over the +total volume (times ``h=\eta+H-H_b`` for the vertical then integrated over the sphere +``\iint dA``). ```math -\iint \frac{h}{2}(u^2 + v^2) + gh^2 dA +E = \iint \left[ \int_{H_b}^{\eta}\frac{1}{2}\left(u^2 + v^2 + gz\right)dz \right]dA \\ + = \iint \frac{1}{2}\left(u^2 + v^2 + gh\right)h dA ``` In contrast to the [Mass conservation](@ref) which, with respect to the @@ -130,7 +134,6 @@ space for the global integral. Let us define a `total_energy` function as ```@example analysis using SpeedyWeather function total_energy(u, v, η, model) - h = zero(u) E = zero(u) # allocate grid variable @@ -138,8 +141,8 @@ function total_energy(u, v, η, model) Hb = model.orography.orography g = model.planet.gravity - @. h = η + H - Hb - @. E = h/2*(u^2 + v^2) + g*h^2 + @. h = η + H - Hb # layer thickness between the bottom and free surface + @. E = h/2*(u^2 + v^2) + g*h^2 # vertically-integrated mechanical energy # transform to spectral, take l=m=0 mode at [1] and normalize for mean E_mean = real(spectral(E)[1]) / model.spectral_transform.norm_sphere @@ -147,7 +150,7 @@ end ``` So at the current state of our simulation we have a total energy -(per square meter as we haven't multiplied by the surface area) +(per square meter as we haven't multiplied by the surface area of the planet) ```@example analysis # flat copies for convenience @@ -173,11 +176,18 @@ So the total energy has somewhat changed, it decreased to ```@example analysis TE_later/TE ``` -of its previous value over 10 days. While technically energy should -be conserved in an unforced system, numerically this is rarely -exactly the case. We need some [Horizontal diffusion](@ref diffusion) -for numerical stability and also the time integration is dissipative -due to temporal filtering, see [Time integration](@ref leapfrog). +of its previous value over 10 days. + +!!! note "Energy conservation" + While technically energy should be conserved in an unforced system, + numerically this is rarely exactly the case. We need some + [Horizontal diffusion](@ref diffusion) for numerical stability and also the time + integration is dissipative due to temporal filtering, see + [Time integration](@ref leapfrog). Note that the energy here is inversely + cascading to larger scales, and this makes the dissipation of energy through + [Horizontal diffusion](@ref diffusion) really inefficient, because in spectral space, + standard Laplacian diffusion is proportional to ``k^2`` where ``k`` + is the wavenumber. ## Potential vorticity @@ -187,7 +197,7 @@ Potential vorticity in the shallow water equations is defined as q = \frac{f + \zeta}{h} ``` -with ``f`` the Coriolis parameter, relative vorticity ``\zeta`` +with ``f`` the Coriolis parameter, ``\zeta`` the relative vorticity, and ``h`` the layer thickness as before. We can calculate this conveniently directly on the model grid (whichever you chose) as @@ -221,12 +231,307 @@ plot(q) ## Absolute angular momentum -More to follow ... +Similar to the total mass, in the absence of sources and sinks for momentum, total +absolute angular momentum defined as + +```math +\Lambda = \iint \left(ur + \Omega r^2\right)h dA +``` + +should be conserved (``\partial_t\Lambda = 0``). Here ``u`` is the zonal velocity, +``\Omega`` the angular velocity of the Earth, ``r = a \cos\phi`` the momentum arm, +and ``a`` the radius of Earth. + +Following previous examples, let us define a `total_angular_momentum` function as + +```@example analysis +using SpeedyWeather + +function total_angular_momentum(u, η, model) + h = zero(u) + Λ = zero(u) # allocate grid variable + + H = model.atmosphere.layer_thickness + Hb = model.orography.orography + + r = model.spectral_grid.radius * cos.(model.geometry.lats) # create r on that grid + + @. h = η + H - Hb # layer thickness between the bottom and free surface + @. Λ = (u*r + model.planet.rotation * r^2) * h # vertically-integrated AAM + + # transform to spectral, take l=m=0 mode at [1] and normalize for mean + Λ_mean = real(spectral(Λ)[1]) / model.spectral_transform.norm_sphere +end +``` + +Anytime we stop the simulation, we could get ``\Lambda`` using this function (multiplied +by ``4\pi a^2`` to get total ``\Lambda``). So after some times of integration, we +could get another ``\Lambda``: + +```@example analysis +run!(simulation, period=Day(10)) + +# u, η got updated during run! +Λ_later = total_angular_momentum(u, η, model) +``` +So the total absolute angular momentum could have changed, where +```@example analysis +Λ_later / Λ_previous +``` +is a measure of its change. Similar to energy, in the numerical integration, Λ +is not exactly conserved due to [Horizontal diffusion](@ref diffusion) and bottom +friction. + ## Circulation -More to follow ... +Total circulation is defined as the area-integrated absolute vorticity: + +```math +C = \iint \left(\zeta + f\right) dA +``` + +Following previous fashion, we define a function ``total_circulation`` for this: + +```@example analysis +using SpeedyWeather + +function total_circulation(ζ, model) + f = coriolis(ζ) # create f on that grid + + # transform to spectral, take l=m=0 mode at [1] and normalize for mean + C_mean = real(spectral(ζ .+ f)[1]) / model.spectral_transform.norm_sphere +end +``` + +!!! note "Global-integrated circulation" + Note that the area integral of relative vorticity ``\zeta`` and planetary vorticity + ``f`` over the whole surface of a sphere are exactly zero. Here ``C_mean`` should + be a small number but may not be perfect-zero due to numerical precision. ## Enstrophy -More to follow ... \ No newline at end of file +The total enstrophy is defined as the second-moment of potential vorticity ``q``: + +```math +Q = \iint \frac{1}{2}q^2 dA +``` + +In the absence of source and sink for potential vorticiy, this quantity shoud also +conserves during the integration. + +We define a function ``total_enstrophy`` for this: + +```@example analysis +# flat copies for convenience +function total_enstrophy(ζ, η, model) + h = zero(ζ) + q = zero(ζ) + + # constants used by Speedy model + H = model.atmosphere.layer_thickness + Hb = model.orography.orography + + f = coriolis(u) # create f on that grid + + @. h = η + H - Hb # thickness + @. q = (ζ + f) / h # PV + + # transform to spectral, take l=m=0 mode at [1] and normalize for mean + Q_mean = real(spectral(q.^2./2)[1]) / model.spectral_transform.norm_sphere +end +``` + +Then by evaluting ``Q_mean`` at different time steps, one could check if ``Q`` is +increasing or decreasing. + +!!! note "Less conservative enstrophy" + Note that the turbulent nature of + [shallow-water model](@ref shallow_water_model) (or 2D turbulence) generally cascade + ``Q`` to smaller scales, making it vulnerable to diffusion. As a result, it is + decreasing more quickly than energy in the presence of small but finite + [Horizontal diffusion](@ref diffusion). + +## Put them all together + +All the above global diagnostics can be put together, and get all of them periodically +during the simulation through the _callbacks_ mechanism (todo: we may need a link here). + +So first, we define a function ``global_diagnostics`` to get all the above integrals as: + +```@example analysis +function global_diagnostics(u, v, vor, η, model) + h = zero(u) + q = zero(u) + λ = zero(u) + k = zero(u) + p = zero(u) + + # constants used by Speedy model + Hbar = model.atmosphere.layer_thickness + Hbtm = model.orography.orography + norm = model.spectral_transform.norm_sphere + Reth = model.spectral_grid.radius + Oeth = model.planet.rotation + grav = model.planet.gravity + + ζ = vor / Reth # todo: do we need to scale the vorticity? + r = Reth * cos.(model.geometry.lats) # create r on that grid + f = SpeedyWeather.coriolis(u) # create f on that grid + + @. h = η + Hbar - Hbtm # thickness + @. q = (ζ + f) / h # PV + @. λ = u * r + Oeth * r^2 + @. k = 1/2 * (u^2 + v^2) + @. p = 1/2 * grav * h + + function ∬dA(v, h) # define a global integral + Vwei = zero(h) + @. Vwei = v * h + + # reuse precomputed spectral transform for performance + return real(spectral(Vwei, model.spectral_transform)[1]) / norm + end + + M = ∬dA(q.^0 , h) # mean mass + C = ∬dA(q.^1 , h) # mean circulation + Λ = ∬dA(λ , h) # mean angular momentum + K = ∬dA(k , h) # mean kinetic energy + P = ∬dA(p , h) # mean potential energy + Q = ∬dA(q.^2/2, h) # mean enstrophy + + return M, C, Λ, K, P, Q +end + +function global_diagnostics(diagn::DiagnosticVariables, model::ModelSetup) + u = diagn.layers[1].grid_variables.u_grid + v = diagn.layers[1].grid_variables.v_grid + ζ = diagn.layers[1].grid_variables.vor_grid + η = diagn.surface.pres_grid + + return global_diagnostics(u, v, ζ, η, model) +end +``` + +Then we define the ``AbstractCallback`` and re-write the functions of ``initialize!``, +``callback!`` and ``finish!``: + +```@example analysis +Base.@kwdef mutable struct GlobalDiagnostics <: SpeedyWeather.AbstractCallback + timestep_counter::Int = 0 + + M::Vector{Float64} = [] # mean mass per time step + C::Vector{Float64} = [] # mean circulation per time step + Λ::Vector{Float64} = [] # mean angular momentum per time step + K::Vector{Float64} = [] # mean kinetic energy per time step + P::Vector{Float64} = [] # mean potential energy per time step + Q::Vector{Float64} = [] # mean enstrophy per time step +end + +function SpeedyWeather.initialize!( + callback::GlobalDiagnostics, + progn::PrognosticVariables, + diagn::DiagnosticVariables, + model::ModelSetup, +) + length = progn.clock.n_timesteps+1 # replace with vector of correct length + callback.M = zeros(length) + callback.C = zeros(length) + callback.Λ = zeros(length) + callback.K = zeros(length) + callback.P = zeros(length) + callback.Q = zeros(length) + + M, C, Λ, K, P, Z = global_diagnostics(diagn, model) + + callback.M[1] = M # set initial conditions + callback.C[1] = C # set initial conditions + callback.Λ[1] = Λ # set initial conditions + callback.K[1] = K # set initial conditions + callback.P[1] = P # set initial conditions + callback.Q[1] = Q # set initial conditions + + callback.timestep_counter = 1 # (re)set counter to 1 +end + +function SpeedyWeather.callback!( + callback::GlobalDiagnostics, + progn::PrognosticVariables, + diagn::DiagnosticVariables, + model::ModelSetup, +) + callback.timestep_counter += 1 + i = callback.timestep_counter + + M, C, Λ, K, P, Q = global_diagnostics(diagn, model) + + callback.M[i] = M # set initial conditions + callback.C[i] = C # set initial conditions + callback.Λ[i] = Λ # set initial conditions + callback.K[i] = K # set initial conditions + callback.P[i] = P # set initial conditions + callback.Q[i] = Q # set initial conditions +end + +function SpeedyWeather.finish!( + callback::GlobalDiagnostics, + progn::PrognosticVariables, + diagn::DiagnosticVariables, + model::ModelSetup, +) + ds = NCDataset(joinpath(path, "run_"*id*"/", "global_diagnostics.nc"), "c") + + defVar(ds, "M", callback.M, ("tstep",)) + defVar(ds, "C", callback.C, ("tstep",)) + defVar(ds, "Λ", callback.Λ, ("tstep",)) + defVar(ds, "K", callback.K, ("tstep",)) + defVar(ds, "P", callback.P, ("tstep",)) + defVar(ds, "Q", callback.Q, ("tstep",)) + + close(ds) +end +``` + +Finally, we could get these global diagnostics _every_ time step by: + +```@example analysis +using SpeedyWeather +using NCDatasets + +# preparing the model configurations +trunc = 170 # Typical resolutions are 42, 63, 85, 127, 170, 255, 340, 511, 682, 1023 +path = "~/SWMrun/" +NF = Float64 +id = "T"*string(trunc) + +spectral_grid = SpectralGrid(NF=NF, trunc=trunc, nlev=1, Grid=FullClenshawGrid, dealiasing=3) +orography = NoOrography(spectral_grid) +initial_conditions = ZonalJet() + +output = OutputWriter(spectral_grid, ShallowWater, output_dt=Day(1), path=path, + id=id, NF=NF, output_vars=[:u, :v, :pres, :vor]) + +model = ShallowWaterModel(;spectral_grid, orography, initial_conditions, output=output) +simulation = initialize!(model) +diagnostics_callback = GlobalDiagnostics() +add!(model.callbacks, diagnostics_callback) +run!(simulation, period=Day(30), output=true) +``` + +Then one could check the output file ``global_diagnostics.nc``, or directly use +the random key of the callback ``:callback_Q3bN`` as: + +```@example analysis +using PythonPlot + +Q = model.callbacks[:callback_Q3bN].Q # get enstrophy + +PythonPlot.plot(Q) +xlabel("time step") +ylabel("total enstrophy") +tight_layout() +``` + +Here is a example plot of all the diagnostics above: +![Overview of implemented grids in SpeedyWeather.jl](https://raw.githubusercontent.com/SpeedyWeather/SpeedyWeather.jl/main/docs/img/diagnostics.png) + From eaf566a8577c7282387ba77e75dae83b148fe770 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 17:27:51 -0400 Subject: [PATCH 02/15] global diagnostics plot created in docs --- docs/src/analysis.md | 318 ++++++++++++++++++++++++++----------------- 1 file changed, 190 insertions(+), 128 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 17843dad0..222377846 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -37,7 +37,8 @@ spectral transforms to reduce the discretization error. The following contains a (hopefully growing) list of examples of how a simulation can be analysed interactively. We call this _online_ analysis because you are directly using the functionality from -the model as if it was a library. For simplicity, we use [shallow-water model](@ref shallow_water_model) to demonstrate this. +the model as if it was a library. For simplicity, we use the +[shallow water model](@ref shallow_water_model) to demonstrate this. ## Mass conservation @@ -95,8 +96,9 @@ So the initial conditions in this simulation are such that the global mean inter is that value in meters. You would need to multiply by the area of the sphere ``4\pi r^2`` (radius ``r``) to get the actual integral from above, but because that doesn't change with time either, we just want to check that `η_mean` doesn't change with time. -Which is equivalent to ``\partial_t \iint \eta dA = 0`` and so volume conservation and because density is constant -also mass conservation. Let's check what happens after the simulation ran for some days +Which is equivalent to ``\partial_t \iint \eta dA = 0`` and so volume conservation and +because density is constant also mass conservation. +Let's check what happens after the simulation ran for some days ```@example analysis model.feedback.verbose = false # hide @@ -110,13 +112,13 @@ which is _exactly_ the same. So mass is conserved, woohoo. Insight from a numerical perspective: The tendency of ``\eta`` is ``\partial_t \eta = -\nabla \cdot (\mathbf{u} h)`` which is a divergence of a flux. -Calculating the divergence in spherical harmonics always sets the ``l=m=0`` mode to zero exactly -(the gradient of a periodic variable has zero mean) so the time integration here is always with -an exactly zero tendency. +Calculating the divergence in spherical harmonics always sets the ``l=m=0`` mode +to zero exactly (the gradient of a periodic variable has zero mean) so the +time integration here is always with an exactly zero tendency. ## Energy -The total energy in the shallow water equation is the sum of the kinetic energy density +The total energy in the [shallow water equations](@ref shallow_water_model) is the sum of the kinetic energy density ``\frac{1}{2}(u^2 + v^2)`` and the potential energy density ``gz`` integrated over the total volume (times ``h=\eta+H-H_b`` for the vertical then integrated over the sphere ``\iint dA``). @@ -128,8 +130,8 @@ E = \iint \left[ \int_{H_b}^{\eta}\frac{1}{2}\left(u^2 + v^2 + gz\right)dz \righ In contrast to the [Mass conservation](@ref) which, with respect to the spectral transform, is a linear calculation, here we need to multiply -variables, which is done in grid-point space. Then we can transform to spectral -space for the global integral. Let us define a `total_energy` function as +variables, which has to be done in grid-point space. Then we can transform to spectral +space for the global integral as before. Let us define a `total_energy` function as ```@example analysis using SpeedyWeather @@ -150,7 +152,7 @@ end ``` So at the current state of our simulation we have a total energy -(per square meter as we haven't multiplied by the surface area of the planet) +(per square meter as we haven't multiplied by the surface area of the planet). ```@example analysis # flat copies for convenience @@ -163,6 +165,8 @@ TE = total_energy(u, v, η, model) with units of ``m^3 s^{-2}`` (multiplying by surface area of the sphere and density of the fluid would turn it into joule = ``kg m^2 s^{-2}``). +To know in general where to find the respective variables ``u, v, \eta`` inside our +simulation object see [Prognostic variables](@ref) and [Diagnostic variables](@ref). Now let us continue the simulation ```@example analysis run!(simulation, period=Day(10)) @@ -187,11 +191,13 @@ of its previous value over 10 days. cascading to larger scales, and this makes the dissipation of energy through [Horizontal diffusion](@ref diffusion) really inefficient, because in spectral space, standard Laplacian diffusion is proportional to ``k^2`` where ``k`` - is the wavenumber. + is the wavenumber. By default, we use a 4th power Laplacian, so the diffusion here + is proportional to ``k^8``. ## Potential vorticity -Potential vorticity in the shallow water equations is defined as +Potential vorticity in the [shallow water equations](@ref shallow_water_model) +is defined as ```math q = \frac{f + \zeta}{h} @@ -232,7 +238,7 @@ plot(q) ## Absolute angular momentum Similar to the total mass, in the absence of sources and sinks for momentum, total -absolute angular momentum defined as +absolute angular momentum (AAM) defined as ```math \Lambda = \iint \left(ur + \Omega r^2\right)h dA @@ -253,34 +259,35 @@ function total_angular_momentum(u, η, model) H = model.atmosphere.layer_thickness Hb = model.orography.orography + a = model.planet.radius + Ω = model.planet.rotation - r = model.spectral_grid.radius * cos.(model.geometry.lats) # create r on that grid + r = a * cos.(model.geometry.lats) # create r on that grid - @. h = η + H - Hb # layer thickness between the bottom and free surface - @. Λ = (u*r + model.planet.rotation * r^2) * h # vertically-integrated AAM + @. h = η + H - Hb # layer thickness between the bottom and free surface + @. Λ = (u*r + Ω*r^2) * h # vertically-integrated AAM # transform to spectral, take l=m=0 mode at [1] and normalize for mean Λ_mean = real(spectral(Λ)[1]) / model.spectral_transform.norm_sphere end ``` -Anytime we stop the simulation, we could get ``\Lambda`` using this function (multiplied -by ``4\pi a^2`` to get total ``\Lambda``). So after some times of integration, we -could get another ``\Lambda``: - +Anytime we stop the simulation, we can calculate ``\Lambda`` using this function +(ignoring the multiplication by ``4\pi a^2`` to get total ``\Lambda``). +```@example analysis +# use u, η from current state of simulation +Λ_current = total_angular_momentum(u, η, model) +``` +So after some days of integration, we would get another ``\Lambda`` with ```@example analysis run!(simulation, period=Day(10)) # u, η got updated during run! Λ_later = total_angular_momentum(u, η, model) +Λ_later / Λ_current ``` -So the total absolute angular momentum could have changed, where -```@example analysis -Λ_later / Λ_previous -``` -is a measure of its change. Similar to energy, in the numerical integration, Λ -is not exactly conserved due to [Horizontal diffusion](@ref diffusion) and bottom -friction. +and measure its relative change. Similar to energy, in the numerical integration, Λ +is not exactly conserved due to [Horizontal diffusion](@ref diffusion). ## Circulation @@ -294,8 +301,6 @@ C = \iint \left(\zeta + f\right) dA Following previous fashion, we define a function ``total_circulation`` for this: ```@example analysis -using SpeedyWeather - function total_circulation(ζ, model) f = coriolis(ζ) # create f on that grid @@ -306,27 +311,28 @@ end !!! note "Global-integrated circulation" Note that the area integral of relative vorticity ``\zeta`` and planetary vorticity - ``f`` over the whole surface of a sphere are exactly zero. Here ``C_mean`` should - be a small number but may not be perfect-zero due to numerical precision. + ``f`` over the whole surface of a sphere are analytically exactly zero. + Numerically, ``C_mean`` should be a small number but may not be exactly zero due + to numerical precision and errors in the spectral transform. -## Enstrophy +## Potential enstrophy -The total enstrophy is defined as the second-moment of potential vorticity ``q``: +The total potential enstrophy is defined as the second-moment of potential vorticity ``q`` ```math Q = \iint \frac{1}{2}q^2 dA ``` -In the absence of source and sink for potential vorticiy, this quantity shoud also -conserves during the integration. +In the absence of source and sink for potential vorticiy, +this quantity should also conserve during the integration. -We define a function ``total_enstrophy`` for this: +We define a function ``total_enstrophy`` for this ```@example analysis -# flat copies for convenience function total_enstrophy(ζ, η, model) h = zero(ζ) q = zero(ζ) + Q = zero(ζ) # constants used by Speedy model H = model.atmosphere.layer_thickness @@ -334,92 +340,125 @@ function total_enstrophy(ζ, η, model) f = coriolis(u) # create f on that grid - @. h = η + H - Hb # thickness - @. q = (ζ + f) / h # PV - + @. h = η + H - Hb # thickness + @. q = (ζ + f) / h # Potential vorticity + @. Q = q^2 / 2 # Potential enstrophy + # transform to spectral, take l=m=0 mode at [1] and normalize for mean - Q_mean = real(spectral(q.^2./2)[1]) / model.spectral_transform.norm_sphere + Q_mean = real(spectral(Q)[1]) / model.spectral_transform.norm_sphere end ``` -Then by evaluting ``Q_mean`` at different time steps, one could check if ``Q`` is -increasing or decreasing. +Then by evaluting ``Q_mean`` at different time steps, one can similarly +check how ``Q`` is changing over time. + +```@example analysis +Q = total_enstrophy(ζ, η, model) +run!(simulation, period=Day(10)) +Q_later = total_enstrophy(ζ, η, model) +Q_later/Q +``` !!! note "Less conservative enstrophy" - Note that the turbulent nature of - [shallow-water model](@ref shallow_water_model) (or 2D turbulence) generally cascade - ``Q`` to smaller scales, making it vulnerable to diffusion. As a result, it is - decreasing more quickly than energy in the presence of small but finite - [Horizontal diffusion](@ref diffusion). + Note that the turbulent nature of the [shallow water model](@ref shallow_water_model) + (or generally 2D turbulence) cascades enstrophy to smaller scales where it + is removed by [Horizontal diffusion](@ref diffusion) for numerical stability. + As a result, it is decreasing more quickly than energy. -## Put them all together -All the above global diagnostics can be put together, and get all of them periodically -during the simulation through the _callbacks_ mechanism (todo: we may need a link here). +## Online diagnostics -So first, we define a function ``global_diagnostics`` to get all the above integrals as: +Now we want to calculate all the above global diagnostics periodically +during a simulation. For that we will use [Callbacks](@ref), which +let us inject code into a simulation that is executed after every +time step (or at any other scheduled time, see [Schedules]@ref)). + +So we define a function `global_diagnostics` to calculate the +integrals together. We could reuse the +functions like `total_enstrophy` from above but we also want +to show how to global integral ``\iint dV`` can be written more efficiently ```@example analysis -function global_diagnostics(u, v, vor, η, model) - h = zero(u) +# define a global integral, reusing a precomputed SpectralTransform S +# times surface area of sphere omitted +function ∬dA(v, h, S::SpectralTransform) + vh = zero(h) + @. vh = v * h + return real(spectral(vh, S)[1]) / S.norm_sphere +end + +# use SpectralTransform from model +∬dA(v, h, model::ModelSetup) = ∬dA(v, h, model.spectral_transform) +``` +By reusing `model.spectral_transform` we do not have to re-precompute +the spectral tranform on every call to `spectral`. Providing +the spectral transform from model as the 2nd argument simply reuses +a previously precomputed spectral transform which is much faster +and uses less memory. + +Now the `global_diagnostics` function is defined as + +```@example analysis +function global_diagnostics(u, v, ζ, η, model) + h = zero(u) # preallocate q = zero(u) λ = zero(u) k = zero(u) p = zero(u) - # constants used by Speedy model - Hbar = model.atmosphere.layer_thickness - Hbtm = model.orography.orography - norm = model.spectral_transform.norm_sphere - Reth = model.spectral_grid.radius - Oeth = model.planet.rotation - grav = model.planet.gravity - - ζ = vor / Reth # todo: do we need to scale the vorticity? - r = Reth * cos.(model.geometry.lats) # create r on that grid - f = SpeedyWeather.coriolis(u) # create f on that grid - - @. h = η + Hbar - Hbtm # thickness - @. q = (ζ + f) / h # PV - @. λ = u * r + Oeth * r^2 - @. k = 1/2 * (u^2 + v^2) - @. p = 1/2 * grav * h - - function ∬dA(v, h) # define a global integral - Vwei = zero(h) - @. Vwei = v * h - - # reuse precomputed spectral transform for performance - return real(spectral(Vwei, model.spectral_transform)[1]) / norm - end - - M = ∬dA(q.^0 , h) # mean mass - C = ∬dA(q.^1 , h) # mean circulation - Λ = ∬dA(λ , h) # mean angular momentum - K = ∬dA(k , h) # mean kinetic energy - P = ∬dA(p , h) # mean potential energy - Q = ∬dA(q.^2/2, h) # mean enstrophy + # constants from model + H = model.atmosphere.layer_thickness + Hb = model.orography.orography + R = model.spectral_grid.radius + Ω = model.planet.rotation + g = model.planet.gravity + + r = R * cos.(model.geometry.lats) # create r on that grid + f = coriolis(u) # create f on that grid + @. h = η + H - Hb # thickness + @. q = (ζ + f) / h # potential vorticity + @. λ = u * r + Ω * r^2 # angular momentum + @. k = 1/2 * (u^2 + v^2) # kinetic energy + @. p = 1/2 * g * h # potential energy + @. z = q^2/2 # potential enstrophy + + M = ∬dA(1, h, model) # mean mass + C = ∬dA(q, h, model) # mean circulation + Λ = ∬dA(λ, h, model) # mean angular momentum + K = ∬dA(k, h, model) # mean kinetic energy + P = ∬dA(p, h, model) # mean potential energy + Q = ∬dA(z, h, model) # mean potential enstrophy + return M, C, Λ, K, P, Q end +# unpack diagnostic variables and call global_diagnostics from above function global_diagnostics(diagn::DiagnosticVariables, model::ModelSetup) u = diagn.layers[1].grid_variables.u_grid v = diagn.layers[1].grid_variables.v_grid - ζ = diagn.layers[1].grid_variables.vor_grid + ζR = diagn.layers[1].grid_variables.vor_grid η = diagn.surface.pres_grid + # vorticity during simulation is scaled by radius R, unscale here + ζ = zero(ζR) + @. ζ = ζR / diagn.scale[] + return global_diagnostics(u, v, ζ, η, model) end ``` -Then we define the ``AbstractCallback`` and re-write the functions of ``initialize!``, -``callback!`` and ``finish!``: +Then we define a new callback `GlobalDiagnostics` subtype of SpeedyWeather's +``AbstractCallback`` and define new methods of ``initialize!``, +``callback!`` and ``finish!`` for it (see [Callbacks](@ref) for more +details) ```@example analysis +# define a GlobalDiagnostics callback and the fields it needs Base.@kwdef mutable struct GlobalDiagnostics <: SpeedyWeather.AbstractCallback timestep_counter::Int = 0 + time::Vector{DateTime} = [] M::Vector{Float64} = [] # mean mass per time step C::Vector{Float64} = [] # mean circulation per time step Λ::Vector{Float64} = [] # mean angular momentum per time step @@ -428,13 +467,15 @@ Base.@kwdef mutable struct GlobalDiagnostics <: SpeedyWeather.AbstractCallback Q::Vector{Float64} = [] # mean enstrophy per time step end +# define how to initialize a GlobalDiagnostics callback function SpeedyWeather.initialize!( callback::GlobalDiagnostics, progn::PrognosticVariables, diagn::DiagnosticVariables, model::ModelSetup, ) - length = progn.clock.n_timesteps+1 # replace with vector of correct length + # replace with vector of correct length + length = progn.clock.n_timesteps + 1 # +1 for initial conditions callback.M = zeros(length) callback.C = zeros(length) callback.Λ = zeros(length) @@ -444,6 +485,7 @@ function SpeedyWeather.initialize!( M, C, Λ, K, P, Z = global_diagnostics(diagn, model) + callback.time[1] = progn.clock.time callback.M[1] = M # set initial conditions callback.C[1] = C # set initial conditions callback.Λ[1] = Λ # set initial conditions @@ -454,6 +496,7 @@ function SpeedyWeather.initialize!( callback.timestep_counter = 1 # (re)set counter to 1 end +# define what a GlobalDiagnostics callback does on every time step function SpeedyWeather.callback!( callback::GlobalDiagnostics, progn::PrognosticVariables, @@ -465,6 +508,7 @@ function SpeedyWeather.callback!( M, C, Λ, K, P, Q = global_diagnostics(diagn, model) + callback.time[i] = progn.clock.time callback.M[i] = M # set initial conditions callback.C[i] = C # set initial conditions callback.Λ[i] = Λ # set initial conditions @@ -473,65 +517,83 @@ function SpeedyWeather.callback!( callback.Q[i] = Q # set initial conditions end +# define how to finish a GlobalDiagnostics callback after simulation finished function SpeedyWeather.finish!( callback::GlobalDiagnostics, progn::PrognosticVariables, diagn::DiagnosticVariables, model::ModelSetup, ) - ds = NCDataset(joinpath(path, "run_"*id*"/", "global_diagnostics.nc"), "c") + n_timesteps = callback.timestep_counter + + # create a netCDF file in current path + ds = NCDataset(joinpath(pwd(), "global_diagnostics.nc"), "c") - defVar(ds, "M", callback.M, ("tstep",)) - defVar(ds, "C", callback.C, ("tstep",)) - defVar(ds, "Λ", callback.Λ, ("tstep",)) - defVar(ds, "K", callback.K, ("tstep",)) - defVar(ds, "P", callback.P, ("tstep",)) - defVar(ds, "Q", callback.Q, ("tstep",)) + defDim(ds, "time", n_timesteps) + defVar(ds, "time", callback,time, ("time",)) + defVar(ds, "mass", callback.M, ("time",)) + defVar(ds, "circulation", callback.C, ("time",)) + defVar(ds, "angular momentum", callback.Λ, ("time",)) + defVar(ds, "kinetic energy", callback.K, ("time",)) + defVar(ds, "potential energy", callback.P, ("time",)) + defVar(ds, "potential enstrophy", callback.Q, ("time",)) close(ds) end ``` -Finally, we could get these global diagnostics _every_ time step by: - -```@example analysis -using SpeedyWeather -using NCDatasets - -# preparing the model configurations -trunc = 170 # Typical resolutions are 42, 63, 85, 127, 170, 255, 340, 511, 682, 1023 -path = "~/SWMrun/" -NF = Float64 -id = "T"*string(trunc) - -spectral_grid = SpectralGrid(NF=NF, trunc=trunc, nlev=1, Grid=FullClenshawGrid, dealiasing=3) -orography = NoOrography(spectral_grid) -initial_conditions = ZonalJet() +Note that `callback!` will execute _every_ time step. If +execution is only desired periodically, you can use [Schedules](@ref). +At `finish!` we decide to write the timeseries of our global +diagnostics as netCDF file via NCDatasets.jl to the current +path `pwd()`. -output = OutputWriter(spectral_grid, ShallowWater, output_dt=Day(1), path=path, - id=id, NF=NF, output_vars=[:u, :v, :pres, :vor]) +Now we create a `GlobalDiagnostics` callback, add it to the +model with key `:global_diagnostics` (you get a random key if not +provided) and reinitialize the simulation to start from the +initial conditions. -model = ShallowWaterModel(;spectral_grid, orography, initial_conditions, output=output) +```@example analysis +global_diagnostics = GlobalDiagnostics() +add!(model.callbacks, :global_diagnostics => global_diagnostics) simulation = initialize!(model) -diagnostics_callback = GlobalDiagnostics() -add!(model.callbacks, diagnostics_callback) -run!(simulation, period=Day(30), output=true) + +run!(simulation, period=Day(20)) ``` -Then one could check the output file ``global_diagnostics.nc``, or directly use -the random key of the callback ``:callback_Q3bN`` as: +Then one could check the output file `global_diagnostics.nc`, +or directly use the callback through its key `:global_diagnostics` +as we do here ```@example analysis using PythonPlot -Q = model.callbacks[:callback_Q3bN].Q # get enstrophy +# unpack callback +(; time, M, C, Λ, K, P, Q) = model.callbacks[:global_diagnostics] -PythonPlot.plot(Q) -xlabel("time step") -ylabel("total enstrophy") -tight_layout() -``` +fig, axs = subplots(3, 2, figsize=(6,6), sharex=true) -Here is a example plot of all the diagnostics above: -![Overview of implemented grids in SpeedyWeather.jl](https://raw.githubusercontent.com/SpeedyWeather/SpeedyWeather.jl/main/docs/img/diagnostics.png) +axs[1,1].plot(time, M) +axs[1,1].set_title("Mass") +axs[2,1].plot(time, Λ) +axs[2,1].set_title("Angular momentum") + +axs[3,1].plot(time, P) +axs[3,1].set_title("Potential energy") +axs[3,1].set_xlabel("time") + +axs[1,2].plot(time, C) +axs[1,2].set_title("Circulation") + +axs[2,2].plot(time, K) +axs[2,2].set_title("Kinetic energy") + +axs[3,2].plot(time, Q) +axs[3,2].set_title("Potential enstrophy") +axs[3,2].set_xlabel("time") + +tight_layout() # hide +savefig("global_diagnostics.png") # hide +``` +![Global diagnostics](global_diagnostics.png) \ No newline at end of file From 2804be19ca21ae75700668722cdebc2357bc8ae3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 17:43:23 -0400 Subject: [PATCH 03/15] delete diagnostics.png --- docs/img/diagnostics.png | Bin 182276 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 docs/img/diagnostics.png diff --git a/docs/img/diagnostics.png b/docs/img/diagnostics.png deleted file mode 100644 index 10dcf3253dbcb48d1bb43045ec71cd1007455797..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 182276 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zb;Q=eAt5$v8xX%#P1}73a8D*2K^2*qnekn~J}rQx$I!e7wG|Dz3csM=uZ6U_YAidV3!Qz zi1xd_(!U{flMlVJ+1q)u-8p$J(9alU(e2paB*H8JuLZa<0d;F3lO`0N=>Z^)UgF0c zD!zU*gpywlwo^<0`a)5B-72bc2G}GGpIw?ACXFhiiDI|VbwiPcdp#~Y)40j#o~1DGSEBHQG<-|Bp}$mYN{h90@Y4o0TsIY3EUnOm$)ha z;5nwiVR(<6-SDh1`Ta?w3p%;`jkKwZ1hH5{l0DsSO1m~biPMM^y)oDq02km@Hsu;K zB-qRm+3a6jF-1K2uT;bTKYsOd$2ImV9wW{~7tj^UURJu4ebMmt{{ymL-u(ao From 658f70f1a044ba59f8437c6ad133441cf844debc Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 17:46:08 -0400 Subject: [PATCH 04/15] spectral_grid.radius not planet.radius --- docs/src/analysis.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 222377846..648dddedc 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -245,8 +245,8 @@ absolute angular momentum (AAM) defined as ``` should be conserved (``\partial_t\Lambda = 0``). Here ``u`` is the zonal velocity, -``\Omega`` the angular velocity of the Earth, ``r = a \cos\phi`` the momentum arm, -and ``a`` the radius of Earth. +``\Omega`` the angular velocity of the Earth, ``r = R \cos\phi`` the momentum arm, +and ``R`` the radius of Earth. Following previous examples, let us define a `total_angular_momentum` function as @@ -259,10 +259,10 @@ function total_angular_momentum(u, η, model) H = model.atmosphere.layer_thickness Hb = model.orography.orography - a = model.planet.radius + R = model.spectral_grid.radius Ω = model.planet.rotation - r = a * cos.(model.geometry.lats) # create r on that grid + r = R * cos.(model.geometry.lats) # create r on that grid @. h = η + H - Hb # layer thickness between the bottom and free surface @. Λ = (u*r + Ω*r^2) * h # vertically-integrated AAM From 93bd766309205135a126825718b750049b54f4b3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 18:03:30 -0400 Subject: [PATCH 05/15] rename callback instance to diagnostics_recorder --- docs/src/analysis.md | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 648dddedc..a3b59dcb8 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -554,22 +554,23 @@ provided) and reinitialize the simulation to start from the initial conditions. ```@example analysis -global_diagnostics = GlobalDiagnostics() -add!(model.callbacks, :global_diagnostics => global_diagnostics) +# don't name it global_diagnostics because that's a function already! +diagnostics_recorder = GlobalDiagnostics() +add!(model.callbacks, :diagnostics_recorder => diagnostics_recorder) simulation = initialize!(model) run!(simulation, period=Day(20)) ``` Then one could check the output file `global_diagnostics.nc`, -or directly use the callback through its key `:global_diagnostics` +or directly use the callback through its key `:diagnostics_recorder` as we do here ```@example analysis using PythonPlot # unpack callback -(; time, M, C, Λ, K, P, Q) = model.callbacks[:global_diagnostics] +(; time, M, C, Λ, K, P, Q) = model.callbacks[:diagnostics_recorder] fig, axs = subplots(3, 2, figsize=(6,6), sharex=true) From 7392457c3b1b74fdc6107ec7f2236fd0d35b04d1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 18:16:06 -0400 Subject: [PATCH 06/15] z wasn't allocated --- docs/src/analysis.md | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index a3b59dcb8..f625279d7 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -400,12 +400,13 @@ Now the `global_diagnostics` function is defined as ```@example analysis function global_diagnostics(u, v, ζ, η, model) - h = zero(u) # preallocate + h = zero(u) # preallocate grids of same size q = zero(u) λ = zero(u) k = zero(u) p = zero(u) - + z = zero(u) + # constants from model H = model.atmosphere.layer_thickness Hb = model.orography.orography @@ -483,7 +484,7 @@ function SpeedyWeather.initialize!( callback.P = zeros(length) callback.Q = zeros(length) - M, C, Λ, K, P, Z = global_diagnostics(diagn, model) + M, C, Λ, K, P, Q = global_diagnostics(diagn, model) callback.time[1] = progn.clock.time callback.M[1] = M # set initial conditions From abc537d7b5887ae5af13c0979252363f993ced30 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 18:33:55 -0400 Subject: [PATCH 07/15] callback.time was not initialized, using NCDatasets --- docs/src/analysis.md | 19 +++++++++++-------- 1 file changed, 11 insertions(+), 8 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index f625279d7..4deaab2ff 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -476,13 +476,14 @@ function SpeedyWeather.initialize!( model::ModelSetup, ) # replace with vector of correct length - length = progn.clock.n_timesteps + 1 # +1 for initial conditions - callback.M = zeros(length) - callback.C = zeros(length) - callback.Λ = zeros(length) - callback.K = zeros(length) - callback.P = zeros(length) - callback.Q = zeros(length) + n = progn.clock.n_timesteps + 1 # +1 for initial conditions + callback.time = zeros(DateTime, n) + callback.M = zeros(n) + callback.C = zeros(n) + callback.Λ = zeros(n) + callback.K = zeros(n) + callback.P = zeros(n) + callback.Q = zeros(n) M, C, Λ, K, P, Q = global_diagnostics(diagn, model) @@ -518,6 +519,8 @@ function SpeedyWeather.callback!( callback.Q[i] = Q # set initial conditions end +using NCDatasets + # define how to finish a GlobalDiagnostics callback after simulation finished function SpeedyWeather.finish!( callback::GlobalDiagnostics, @@ -531,7 +534,7 @@ function SpeedyWeather.finish!( ds = NCDataset(joinpath(pwd(), "global_diagnostics.nc"), "c") defDim(ds, "time", n_timesteps) - defVar(ds, "time", callback,time, ("time",)) + defVar(ds, "time", callback.time, ("time",)) defVar(ds, "mass", callback.M, ("time",)) defVar(ds, "circulation", callback.C, ("time",)) defVar(ds, "angular momentum", callback.Λ, ("time",)) From 488c74be45a5dbaefaa5005f6d42e8e6ba035311 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 18:35:57 -0400 Subject: [PATCH 08/15] python indexing for axes --- docs/src/analysis.md | 29 +++++++++++++++-------------- 1 file changed, 15 insertions(+), 14 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 4deaab2ff..045391a19 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -578,25 +578,26 @@ using PythonPlot fig, axs = subplots(3, 2, figsize=(6,6), sharex=true) -axs[1,1].plot(time, M) -axs[1,1].set_title("Mass") +# note: python indexing +axs[0,0].plot(time, M) +axs[0,0].set_title("Mass") -axs[2,1].plot(time, Λ) -axs[2,1].set_title("Angular momentum") +axs[1,0].plot(time, Λ) +axs[1,0].set_title("Angular momentum") -axs[3,1].plot(time, P) -axs[3,1].set_title("Potential energy") -axs[3,1].set_xlabel("time") +axs[2,0].plot(time, P) +axs[2,0].set_title("Potential energy") +axs[2,0].set_xlabel("time") -axs[1,2].plot(time, C) -axs[1,2].set_title("Circulation") +axs[0,1].plot(time, C) +axs[0,1].set_title("Circulation") -axs[2,2].plot(time, K) -axs[2,2].set_title("Kinetic energy") +axs[1,1].plot(time, K) +axs[1,1].set_title("Kinetic energy") -axs[3,2].plot(time, Q) -axs[3,2].set_title("Potential enstrophy") -axs[3,2].set_xlabel("time") +axs[2,1].plot(time, Q) +axs[2,1].set_title("Potential enstrophy") +axs[2,1].set_xlabel("time") tight_layout() # hide savefig("global_diagnostics.png") # hide From 5b1704b44ab394cbd0acd99791c3eca90eedbc96 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 18:38:58 -0400 Subject: [PATCH 09/15] avoid naming conflict with time --- docs/src/analysis.md | 15 ++++++++------- 1 file changed, 8 insertions(+), 7 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 045391a19..906670406 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -574,28 +574,29 @@ as we do here using PythonPlot # unpack callback -(; time, M, C, Λ, K, P, Q) = model.callbacks[:diagnostics_recorder] +(; M, C, Λ, K, P, Q) = model.callbacks[:diagnostics_recorder] +t = model.callbacks[:diagnostics_recorder].time fig, axs = subplots(3, 2, figsize=(6,6), sharex=true) # note: python indexing -axs[0,0].plot(time, M) +axs[0,0].plot(t, M) axs[0,0].set_title("Mass") -axs[1,0].plot(time, Λ) +axs[1,0].plot(t, Λ) axs[1,0].set_title("Angular momentum") -axs[2,0].plot(time, P) +axs[2,0].plot(t, P) axs[2,0].set_title("Potential energy") axs[2,0].set_xlabel("time") -axs[0,1].plot(time, C) +axs[0,1].plot(t, C) axs[0,1].set_title("Circulation") -axs[1,1].plot(time, K) +axs[1,1].plot(t, K) axs[1,1].set_title("Kinetic energy") -axs[2,1].plot(time, Q) +axs[2,1].plot(t, Q) axs[2,1].set_title("Potential enstrophy") axs[2,1].set_xlabel("time") From 53e68abbb466228666ebd5c68aea39f7f5d5adb2 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Thu, 28 Mar 2024 18:51:40 -0400 Subject: [PATCH 10/15] time axis format --- docs/src/analysis.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 906670406..c0b4ec71f 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -577,7 +577,7 @@ using PythonPlot (; M, C, Λ, K, P, Q) = model.callbacks[:diagnostics_recorder] t = model.callbacks[:diagnostics_recorder].time -fig, axs = subplots(3, 2, figsize=(6,6), sharex=true) +fig, axs = subplots(3, 2, figsize=(8,6), sharex=true) # note: python indexing axs[0,0].plot(t, M) @@ -600,6 +600,10 @@ axs[2,1].plot(t, Q) axs[2,1].set_title("Potential enstrophy") axs[2,1].set_xlabel("time") +# format time axis to Jan-1 etc +time_fmt = matplotlib.dates.DateFormatter("%b-%d") +axs[0,0].xaxis.set_major_formatter(time_fmt) + tight_layout() # hide savefig("global_diagnostics.png") # hide ``` From 54e779b6258545965d7bd1db55f9f79b16f0ed49 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Fri, 29 Mar 2024 11:18:43 -0400 Subject: [PATCH 11/15] Analysis docs: apply Navid's suggestions Co-authored-by: Navid C. Constantinou --- docs/src/analysis.md | 38 +++++++++++++++++++++++--------------- 1 file changed, 23 insertions(+), 15 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index c0b4ec71f..1666fbc87 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -124,8 +124,10 @@ total volume (times ``h=\eta+H-H_b`` for the vertical then integrated over the s ``\iint dA``). ```math -E = \iint \left[ \int_{H_b}^{\eta}\frac{1}{2}\left(u^2 + v^2 + gz\right)dz \right]dA \\ - = \iint \frac{1}{2}\left(u^2 + v^2 + gh\right)h dA +\begin{align} +E & = \iint \left[ \int_{H_b}^{\eta}\frac{1}{2}\left(u^2 + v^2 + gz\right)dz \right]dA \\ + & = \iint \frac{1}{2}\left(u^2 + v^2 + gh\right)h dA +\end{align} ``` In contrast to the [Mass conservation](@ref) which, with respect to the @@ -136,8 +138,9 @@ space for the global integral as before. Let us define a `total_energy` function ```@example analysis using SpeedyWeather function total_energy(u, v, η, model) + # allocate grid variable h = zero(u) - E = zero(u) # allocate grid variable + E = zero(u) H = model.atmosphere.layer_thickness Hb = model.orography.orography @@ -147,7 +150,7 @@ function total_energy(u, v, η, model) @. E = h/2*(u^2 + v^2) + g*h^2 # vertically-integrated mechanical energy # transform to spectral, take l=m=0 mode at [1] and normalize for mean - E_mean = real(spectral(E)[1]) / model.spectral_transform.norm_sphere + return E_mean = real(spectral(E)[1]) / model.spectral_transform.norm_sphere end ``` @@ -164,7 +167,7 @@ TE = total_energy(u, v, η, model) ``` with units of ``m^3 s^{-2}`` (multiplying by surface area of the sphere -and density of the fluid would turn it into joule = ``kg m^2 s^{-2}``). +and density of the fluid would turn it into joule = ``kg \, m^2 s^{-2}``). To know in general where to find the respective variables ``u, v, \eta`` inside our simulation object see [Prognostic variables](@ref) and [Diagnostic variables](@ref). Now let us continue the simulation @@ -254,8 +257,9 @@ Following previous examples, let us define a `total_angular_momentum` function a using SpeedyWeather function total_angular_momentum(u, η, model) + # allocate grid variable h = zero(u) - Λ = zero(u) # allocate grid variable + Λ = zero(u) H = model.atmosphere.layer_thickness Hb = model.orography.orography @@ -268,7 +272,7 @@ function total_angular_momentum(u, η, model) @. Λ = (u*r + Ω*r^2) * h # vertically-integrated AAM # transform to spectral, take l=m=0 mode at [1] and normalize for mean - Λ_mean = real(spectral(Λ)[1]) / model.spectral_transform.norm_sphere + return Λ_mean = real(spectral(Λ)[1]) / model.spectral_transform.norm_sphere end ``` @@ -298,21 +302,21 @@ Total circulation is defined as the area-integrated absolute vorticity: C = \iint \left(\zeta + f\right) dA ``` -Following previous fashion, we define a function ``total_circulation`` for this: +Following previous fashion, we define a function `total_circulation` for this: ```@example analysis function total_circulation(ζ, model) f = coriolis(ζ) # create f on that grid # transform to spectral, take l=m=0 mode at [1] and normalize for mean - C_mean = real(spectral(ζ .+ f)[1]) / model.spectral_transform.norm_sphere + return C_mean = real(spectral(ζ .+ f)[1]) / model.spectral_transform.norm_sphere end ``` !!! note "Global-integrated circulation" Note that the area integral of relative vorticity ``\zeta`` and planetary vorticity ``f`` over the whole surface of a sphere are analytically exactly zero. - Numerically, ``C_mean`` should be a small number but may not be exactly zero due + Numerically, `C_mean` should be a small number but may not be exactly zero due to numerical precision and errors in the spectral transform. ## Potential enstrophy @@ -345,11 +349,11 @@ function total_enstrophy(ζ, η, model) @. Q = q^2 / 2 # Potential enstrophy # transform to spectral, take l=m=0 mode at [1] and normalize for mean - Q_mean = real(spectral(Q)[1]) / model.spectral_transform.norm_sphere + return Q_mean = real(spectral(Q)[1]) / model.spectral_transform.norm_sphere end ``` -Then by evaluting ``Q_mean`` at different time steps, one can similarly +Then by evaluting `Q_mean` at different time steps, one can similarly check how ``Q`` is changing over time. ```@example analysis @@ -371,7 +375,7 @@ Q_later/Q Now we want to calculate all the above global diagnostics periodically during a simulation. For that we will use [Callbacks](@ref), which let us inject code into a simulation that is executed after every -time step (or at any other scheduled time, see [Schedules]@ref)). +time step (or at any other scheduled time, see [Schedules](@ref)). So we define a function `global_diagnostics` to calculate the integrals together. We could reuse the @@ -450,8 +454,8 @@ end ``` Then we define a new callback `GlobalDiagnostics` subtype of SpeedyWeather's -``AbstractCallback`` and define new methods of ``initialize!``, -``callback!`` and ``finish!`` for it (see [Callbacks](@ref) for more +`AbstractCallback` and define new methods of `initialize!`, +`callback!` and `finish!` for it (see [Callbacks](@ref) for more details) ```@example analysis @@ -496,6 +500,8 @@ function SpeedyWeather.initialize!( callback.Q[1] = Q # set initial conditions callback.timestep_counter = 1 # (re)set counter to 1 + + return nothing end # define what a GlobalDiagnostics callback does on every time step @@ -543,6 +549,8 @@ function SpeedyWeather.finish!( defVar(ds, "potential enstrophy", callback.Q, ("time",)) close(ds) + + return nothing end ``` From 60a9c1a192ebb50303bbcc261a24bc58b266c1ad Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Fri, 29 Mar 2024 11:36:36 -0400 Subject: [PATCH 12/15] radius always R, note on scaling --- docs/src/analysis.md | 22 +++++++++++++++++----- 1 file changed, 17 insertions(+), 5 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 1666fbc87..6141bc640 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -94,7 +94,7 @@ a = model.spectral_transform.norm_sphere # = 2√π = 3.5449078 So the initial conditions in this simulation are such that the global mean interface displacement is that value in meters. You would need to multiply by the area of the sphere -``4\pi r^2`` (radius ``r``) to get the actual integral from above, but because that doesn't +``4\pi R^2`` (radius ``R``) to get the actual integral from above, but because that doesn't change with time either, we just want to check that `η_mean` doesn't change with time. Which is equivalent to ``\partial_t \iint \eta dA = 0`` and so volume conservation and because density is constant also mass conservation. @@ -248,7 +248,7 @@ absolute angular momentum (AAM) defined as ``` should be conserved (``\partial_t\Lambda = 0``). Here ``u`` is the zonal velocity, -``\Omega`` the angular velocity of the Earth, ``r = R \cos\phi`` the momentum arm, +``\Omega`` the angular velocity of the Earth, ``r = R \cos\phi`` the momentum arm at latitude ``\phi``, and ``R`` the radius of Earth. Following previous examples, let us define a `total_angular_momentum` function as @@ -266,7 +266,7 @@ function total_angular_momentum(u, η, model) R = model.spectral_grid.radius Ω = model.planet.rotation - r = R * cos.(model.geometry.lats) # create r on that grid + r = R * cos.(model.geometry.lats) # momentum arm for every grid point @. h = η + H - Hb # layer thickness between the bottom and free surface @. Λ = (u*r + Ω*r^2) * h # vertically-integrated AAM @@ -277,7 +277,7 @@ end ``` Anytime we stop the simulation, we can calculate ``\Lambda`` using this function -(ignoring the multiplication by ``4\pi a^2`` to get total ``\Lambda``). +(ignoring the multiplication by ``4\pi R^2`` to get total ``\Lambda``). ```@example analysis # use u, η from current state of simulation Λ_current = total_angular_momentum(u, η, model) @@ -453,6 +453,17 @@ function global_diagnostics(diagn::DiagnosticVariables, model::ModelSetup) end ``` +!!! note "Radius scaling of vorticity and divergence" + The prognostic variables vorticity and divergence are scaled with the radius of the + sphere during a simulation, see [Radius scaling](@ref scaling). This did not apply + above because we only analyzed vorticity _before_ or _after_ the simulation, i.e. + outside of the `run!(simulation)` call. The radius scaling is only applied just + before the time integration and is undone directly after it. However, because + now we are accessing the vorticity _during_ the simulation we need to unscale + the vorticity (and divergence) manually. General recommendation is to divide + by `diagn.scale[]` (and not `radius`) as `diagn.scale[]` always reflects whether + a vorticity and divergence are currently scaled (scale = radius) or not (scale = 1). + Then we define a new callback `GlobalDiagnostics` subtype of SpeedyWeather's `AbstractCallback` and define new methods of `initialize!`, `callback!` and `finish!` for it (see [Callbacks](@ref) for more @@ -558,7 +569,8 @@ Note that `callback!` will execute _every_ time step. If execution is only desired periodically, you can use [Schedules](@ref). At `finish!` we decide to write the timeseries of our global diagnostics as netCDF file via NCDatasets.jl to the current -path `pwd()`. +path `pwd()`. We need to add `using NCDatasets` here, as SpeedyWeather +does not re-export the functionality therein. Now we create a `GlobalDiagnostics` callback, add it to the model with key `:global_diagnostics` (you get a random key if not From 1348bf8ef15e7ae0826ab39f8896c3963e8aded7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Fri, 29 Mar 2024 11:45:34 -0400 Subject: [PATCH 13/15] Add circulation calculation --- docs/src/analysis.md | 10 +++++++--- 1 file changed, 7 insertions(+), 3 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index 6141bc640..dec56be43 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -146,8 +146,8 @@ function total_energy(u, v, η, model) Hb = model.orography.orography g = model.planet.gravity - @. h = η + H - Hb # layer thickness between the bottom and free surface - @. E = h/2*(u^2 + v^2) + g*h^2 # vertically-integrated mechanical energy + @. h = η + H - Hb # layer thickness between the bottom and free surface + @. E = h/2*(u^2 + v^2) + g*h^2 # vertically-integrated mechanical energy # transform to spectral, take l=m=0 mode at [1] and normalize for mean return E_mean = real(spectral(E)[1]) / model.spectral_transform.norm_sphere @@ -302,7 +302,7 @@ Total circulation is defined as the area-integrated absolute vorticity: C = \iint \left(\zeta + f\right) dA ``` -Following previous fashion, we define a function `total_circulation` for this: +Following previous fashion, we define a function `total_circulation` for this ```@example analysis function total_circulation(ζ, model) @@ -311,6 +311,8 @@ function total_circulation(ζ, model) # transform to spectral, take l=m=0 mode at [1] and normalize for mean return C_mean = real(spectral(ζ .+ f)[1]) / model.spectral_transform.norm_sphere end + +total_circulation(ζ, model) ``` !!! note "Global-integrated circulation" @@ -584,6 +586,7 @@ add!(model.callbacks, :diagnostics_recorder => diagnostics_recorder) simulation = initialize!(model) run!(simulation, period=Day(20)) +nothing # hide ``` Then one could check the output file `global_diagnostics.nc`, @@ -626,5 +629,6 @@ axs[0,0].xaxis.set_major_formatter(time_fmt) tight_layout() # hide savefig("global_diagnostics.png") # hide +nothing # hide ``` ![Global diagnostics](global_diagnostics.png) \ No newline at end of file From cce23bd964a5fd548f3947aa17a3e69aeb04e90b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Fri, 29 Mar 2024 11:48:09 -0400 Subject: [PATCH 14/15] remove incorrect "set initial condition" comment --- docs/src/analysis.md | 14 ++++++++------ 1 file changed, 8 insertions(+), 6 deletions(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index dec56be43..cb6d7434e 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -529,13 +529,14 @@ function SpeedyWeather.callback!( M, C, Λ, K, P, Q = global_diagnostics(diagn, model) + # store current time and diagnostics for timestep i callback.time[i] = progn.clock.time - callback.M[i] = M # set initial conditions - callback.C[i] = C # set initial conditions - callback.Λ[i] = Λ # set initial conditions - callback.K[i] = K # set initial conditions - callback.P[i] = P # set initial conditions - callback.Q[i] = Q # set initial conditions + callback.M[i] = M + callback.C[i] = C + callback.Λ[i] = Λ + callback.K[i] = K + callback.P[i] = P + callback.Q[i] = Q end using NCDatasets @@ -552,6 +553,7 @@ function SpeedyWeather.finish!( # create a netCDF file in current path ds = NCDataset(joinpath(pwd(), "global_diagnostics.nc"), "c") + # save diagnostics variables within defDim(ds, "time", n_timesteps) defVar(ds, "time", callback.time, ("time",)) defVar(ds, "mass", callback.M, ("time",)) From eaaa9612e94593844751da5dcbd3a724f9170090 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Milan=20Kl=C3=B6wer?= Date: Fri, 29 Mar 2024 12:03:10 -0400 Subject: [PATCH 15/15] =?UTF-8?q?=CE=B6=20.+=20f=20escaped=20grid?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/src/analysis.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/docs/src/analysis.md b/docs/src/analysis.md index cb6d7434e..e2ca57ea1 100644 --- a/docs/src/analysis.md +++ b/docs/src/analysis.md @@ -307,9 +307,11 @@ Following previous fashion, we define a function `total_circulation` for this ```@example analysis function total_circulation(ζ, model) f = coriolis(ζ) # create f on that grid + C = zero(ζ) + @. C = ζ + f # transform to spectral, take l=m=0 mode at [1] and normalize for mean - return C_mean = real(spectral(ζ .+ f)[1]) / model.spectral_transform.norm_sphere + return C_mean = real(spectral(C)[1]) / model.spectral_transform.norm_sphere end total_circulation(ζ, model)