From 937b5f10d7d16255f5f75316954745bd3870d322 Mon Sep 17 00:00:00 2001 From: Akshay Sridhar Date: Tue, 8 Aug 2023 15:16:53 -0700 Subject: [PATCH] modified: README.md --- experiments/AMIP/modular/README.md | 82 ++++++++++++++++++++---------- 1 file changed, 54 insertions(+), 28 deletions(-) diff --git a/experiments/AMIP/modular/README.md b/experiments/AMIP/modular/README.md index 34a53ef24..5ccd139da 100644 --- a/experiments/AMIP/modular/README.md +++ b/experiments/AMIP/modular/README.md @@ -4,60 +4,81 @@ The momentum equations are in the advective form, and tracers in the consevative form, namely: - Density: -$$ \frac{\partial \rho}{\partial t} + \nabla \cdot ({\rho \vec{u}})= 0 $$ + +$$ +\frac{\partial \rho}{\partial t} + \nabla \cdot ({\rho \vec{u}})= 0 +$$ - Momentum (flux form): -$$ \frac{\partial \vec{u_h}}{\partial t} + \vec{u} \cdot \nabla \vec{u_h} = - \frac{1}{\rho}\nabla_h p -+ \frac{\partial}{\partial z} K_v \frac{\partial}{\partial z} \vec{u_h} + $$ -$$ \frac{\partial w}{\partial t} + \vec{u} \cdot \nabla w= -- \frac{1}{\rho}\frac{\partial p}{\partial z} -- \nabla_z \Phi -+ \frac{\partial}{\partial z} K_v \frac{\partial}{\partial z} w +\frac{\partial \vec{u_h}}{\partial t} + \vec{u} \cdot \nabla \vec{u_h} = - \frac{1}{\rho}\nabla_h p + \frac{\partial}{\partial z} K_v \frac{\partial}{\partial z} \vec{u_h} +$$ + +$$ +\frac{\partial w}{\partial t} + \vec{u} \cdot \nabla w = - \frac{1}{\rho}\frac{\partial p}{\partial z} - \nabla_z \Phi + \frac{\partial}{\partial z} K_v \frac{\partial}{\partial z} w $$ - Total energy: -$$ \frac{\partial \rho e_{tot}}{\partial t} + \nabla \cdot (\rho h_{tot} \vec{u}) = \frac{\partial}{\partial z} K_v \frac{\partial}{\partial z} h_{tot} + +$$ +\frac{\partial \rho e_{tot}}{\partial t} + \nabla \cdot (\rho h_{tot} \vec{u}) = \frac{\partial}{\partial z} K_v \frac{\partial}{\partial z} h_{tot} $$ where the total specific enthalpy and total specific energy are -$$ h_{tot} = e_{tot} + \frac{p}{\rho} \,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\, e_{tot} = c_v T + \Phi + \frac{1}{2}\vec{u}^2 + +$$ +h_{tot} = e_{tot} + \frac{p}{\rho} $$ + +$$ +e_{tot} = c_v T + \Phi + \frac{1}{2}\vec{u}^2 +$$ + (note that $h_{tot} \neq h = c_vT + p/\rho = c_p T$, the specific enthalpy in the thermodynamic sense), $\Phi = gz$ is the geopotential, $u_h$ is the horizontal velocity vector, $w$ the vertical velocity, $\rho$ the density, $p$ pressure, $K_v$ the vertical diffusivity (assumed constant here). ## Boundary conditions (BCs) - We implement BCs similarly to other climate models. - First-order fluxes (i.e., advective fluxes) are always set to zero, corresponding to the *free-slip* and *impenetrable* BC, where: - $$ - w = 0 \,\,\,\,\,\,\, \partial_t w = 0 \,\,\,\,\,\,\, \nabla \times\vec{u_h}=0 \,\,\,\,\,\,\, \nabla \cdot \vec{\rho u_h}=0 \,\,\,\,\,\,\, \nabla \cdot \rho h_{tot} \vec{u_h}=0 - $$ + + $$w = 0$$ + + $$\partial_t w = 0$$ + + $$\nabla \times\vec{u_h}=0 $$ + + $$\nabla \cdot \vec{\rho u_h}=0 $$ + + $$\nabla \cdot \rho h_{tot} \vec{u_h}=0$$ + - Second-order fluxes (i.e., diffusive fluxes) - `No Flux`: By default we have *impenetrable* or *insulating* BCs (no second-order fluxes) at all boundaries. - `Bulk Formula`: Applied to tracers (e.g., temperature and moisture), this imposes a boundary fluxes (e.g., sensible and latent heat) calculated using the bulk aerodynamic formulae using prescribed surface values of ($T_{sfc}$ and $q_{sfc}^{sat}$). At the surface, the bulk sensible heat flux formula for total enthalpy essentially replaces the above: - $$ (K_v \rho \partial_z h_{tot})_{sfc}$$ + + $$(K_v \rho \partial_z h_{tot})_{sfc}$$ + For **total energy**, we have two choices: + - 1. enthalpy flux: - $$ (K_v \rho \partial_z h_{tot})_{sfc} \rightarrow - \hat{n} \cdot \rho C_H ||u||^{1} (h^1- h_{sfc}) - = F_S - $$ + + $$(K_v \rho \partial_z h_{tot})_{sfc} \rightarrow \hat{n} \cdot \rho C_H ||u||^{1} (h^1- h_{sfc}) = F_S$$ + - 2. sensible (and latent) heat flux. The sensible heat flux is: - $$ (K_v \rho \partial_z h_{tot})_{sfc} \rightarrow - \hat{n} \cdot C_H c_{pd} ρ^{1} ||u||^{1} (T^{1} - T_{sfc}) - + \hat{n} \cdot C_H ρ^{1} ||u||^{1} (\Phi^{1} - \Phi_{sfc}) - = F_S + + $$ + (K_v \rho \partial_{z} h_{tot})_{sfc} \rightarrow \hat{n} \cdot C_H c_{pd} ρ^{1} ||u||^{1} (T^{1} - T_{sfc}) + \hat{n} \cdot C_H ρ^{1} ||u||^{1} (\Phi^{1} - \Phi_{sfc}) = F_S $$ + where $^{1}$ corresponds to the lowest model level, $C_H$ is the dimensionless thermal transfer coefficient, $c_{pd}$ is the specific heat capacity for dry air $||u||$ the wind speed. This is the *bulk turbulent sensible heat flux* parameterization, and $F_S$ is positive when atmosphere receives energy from the surface. The contribution of the kinetic energy is usually O(1e4) smaller and is neglected, but it can be added to F_S as: - $$ - F_{S_{tot}} = F_S + \hat{n} \cdot C_D ρ^{1} ||u||^{1} (\vec{u_h}^{1})^2 - $$ + + $$F_{S_{tot}} = F_S + \hat{n} \cdot C_D ρ^{1} ||u||^{1} (\vec{u_h}^{1})^2$$ + - `Drag Law`: essentially the bulk formula for momentum - $$ \frac{\partial}{\partial z} K_v \frac{\partial}{\partial z} \vec{u_h} \rightarrow - \hat{n} \cdot C_D ρ^{1} ||u||^{1} \vec{u_h}^{1} - = F_M - $$ + + $$\frac{\partial}{\partial z} K_v \frac{\partial}{\partial z} \vec{u_h} \rightarrow \hat{n} \cdot C_D ρ^{1} ||u||^{1} \vec{u_h}^{1} = F_M$$ + - `Coupled Bulk Formula`: same as `Bulk Formula`, but surface quantities (e.g. $T_{sfc}$) are passed from the state of the neighboring model. - The diffusive fluxes are applied via the `vertical_diffusion` ClimaAtmos model sub-component. To apply boundary fluxes without diffusion in the atmospheric interior, the viscosity coefficient needs to be set to zero: $ν = FT(0)$. @@ -83,6 +104,7 @@ https://climate.ucdavis.edu/pubs/UMJS2013QJRMS.pdf # Heat Slab The slab solves for temperature in a single layer, whose tendency is the accumulated fluxes divided by the coupling timestep plus a parameterisation of the internal processes, $G$. + $$ \rho c h_s \, \partial_t T_{sfc} = - F_{integ} / \Delta t_{coupler} $$ @@ -126,13 +148,17 @@ $$ # Prescribed SST and Sea Ice - We simply prescribe SSTs from a file as `T_sfc`. As for sea ice, we will follow GFDL's [AMIP setup](https://pcmdi.llnl.gov/mips/amip/home/Documentation/20gfdl.html#RTFToC31) and use prescribed sea ice concentrations and a constant ice thickness, $h_{i} = 2m$ ice thickness, while solving for $T_{sfc}$: + $$ \frac{dT_{sfc}}{dt} = - \frac{h_i(F_{atm} - F_{conductive})}{k_i} $$ + where + $$ F_{conductive} = \frac{k_i (T_{base} - {T_{sfc}})}{h_{i}} $$ + with the thermal conductivity of ice, $k_i = 2$ W m$^{-2}$ K$^{-1}$, and $T_{base} = 273.16$ K. For now we use an Euler timestepper (and use $T_{sfc}$ of the previous timestep), though this may be solved implicitly in the future. ## Data source