modified: README.md

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