After these generalizations, it makes sense to consider which basic approximations are still made in the dynamic core developed here:
This section is about determining variables that are not direct model variables but are expected by weather models as standard.
The so-called 2-m-temperature is the air temperature 2 m above the earth's surface. According to the WMO standard, the temperature near the ground is measured there and must therefore also be written out by models. To calculate it, one starts from the heat conduction equation Eq. (5.221)
\[ \begin{align} \frac{\partial\psi}{\partial t} = \kappa\frac{\partial^2\psi}{\partial z^2} \end{align} \]
in one-dimensional form with $\kappa > 0$ as the diffusion coefficient. This can now be solved for $z>0$ under the boundary condition
\[ \begin{align} \psi\left(0,t\right) = \psi_0\exp\left(-i\omega t\right). \end{align} \]
You now make the approach
\[ \begin{align} \psi\left(z,t\right) = A\exp\left(ikz - i\omega t\right), \end{align} \]
this implies
\[ \begin{align} -i\omega = -k^2\kappa \Rightarrow k = \pm\sqrt{i}\sqrt{\frac{\omega}{\kappa}} = \pm\sqrt{\frac{\omega}{2\kappa}}\left(1 + i\right). \end{align} \]
The solution with a negative sign can be physically excluded because it leads to an exponentially increasing amplitude. So the solution is
\[ \begin{align} \psi\left(z,t\right) &= \psi_0\exp\left(-i\omega t\right)\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right)\exp\left(i\sqrt{\frac{\omega}{2\kappa}}z\right)\nonumber\\ &= \psi\left(0,t\right)\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right)\exp\left(i\sqrt{\frac{\omega}{2\kappa}}z\right). \end{align} \]
Now apply this solution to the problem of determining the 2 m temperature, i.e. $\psi \to T$. For $\omega$ you use $\omega = \frac{2\pi}{24\:\text{h}}$, i.e. the angular frequency of the daily temperature cycle. The phase component can be neglected as a first approximation. For $\kappa$ one uses the vertical turbulent diffusion coefficient. The solution thus becomes
\[ \begin{align} T\left(z\right) = T_1 + T_2\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right), \end{align} \]
where o. b. d. A. an offset $T_1$ was added and $\psi_0$ was renamed to $T_2$. The boundary conditions apply to this in the atmosphere
\[ \begin{align} T\left(0\right) &= T_1 + T_2,\\ T\left(z_l\right) &= T_1 + T_2\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right), \end{align} \]
Here $T\left(0\right) =: T_l$ is the temperature of the earth's surface and $T\left(z_l\right) =: T_l$ is the temperature of the lowest model layer above ground. From this you get
\[ \begin{align} T_1 &= T_s - T_2,\\ T_l &= T_s - T_2 + T_2\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right) = T_s - T_2\left[1 - \exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right)\right] \Rightarrow T_2 = \frac{T_s - T_l}{1 - \exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right)}. \end{align} \]
It follows
\[ \begin{align} T\left(z\right) &= T_s - \frac{T_s - T_l}{1 - \exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right)} + \frac{T_s - T_l}{1 - \exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right)}\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right)\nonumber\\ &= T_s + \left(T_l - T_s\right)\frac{1 - \exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right)}{1 - \exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right)}. \end{align} \]
From this the 2 m temperature can be determined with $z = 2\:\text{m}$.
The wind is measured by default ten meters above ground, where the vectors of a model layer are usually not located. For this purpose, the logarithmic wind profile Eq. is used. (17.43), so
\[ \begin{align} v_h\left(z_{10}\right) &= \frac{u_\star}{k}\ln\left(\frac{z_{10}}{z_0}\right),\\ v_h\left(z_l\right) &= \frac{u_\star}{k}\ln\left(\frac{z_l}{z_0}\right). \end{align} \]
Here $v_h\geq 0$ is the horizontal wind speed, $z_l$ is the height of the lowest model layer above ground and $z_{10} \coloneqq 10\:\text{m}$. From this it follows
\[ \begin{align} v_h\left(z_{10}\right) &= v_h\left(z_l\right)\frac{\ln\left(\frac{z_{10}}{z_0}\right)}{\ln\left(\frac{z_l}{z_0}\right)}. \end{align} \]
This only applies over water. Over land, the convention is that the wind must be measured over grass in order to establish comparability. The equations are used for this
\[ \begin{align} v_h\left(z_{10}\right) &= \frac{u_\star}{k}\ln\left(\frac{z_{10}}{z_g}\right),\\ v_h\left(z_l\right) &= \frac{u_\star}{k}\ln\left(\frac{z_l}{z_0}\right), \end{align} \]
where $z_g$ is the roughness length of grass (0.02 m) and the friction velocity $u_\star$ was assumed to be the same in both cases. This leads to
\[ \begin{align} v_h\left(z_{10}\right) &= v_h\left(z_l\right)\frac{\ln\left(\frac{z_{10}}{z_g}\right)}{\ln\left(\frac{z_l}{z_0}\right)}. \end{align} \]
The total cloud cover (English total cloud cover (TCC)) is the proportion of the sky covered by clouds. The correct way to calculate the TCC is therefore to form an integral over the half-space and relate the cloud-covered portion to the total portion ($2\pi$). However, a simplified approach is usually chosen in models. Let $N_L$ be the number of layers and $c_{f,i}$ be the cloud-filled portion of the $i-$th layer.
The first step is to calculate from $c_{f,i}$ the non-cloud-covered vertical area fraction $f_i$ of the grid box. Assuming that the clouds consist of a single layer of thickness $d_i$, one obtains
\[ \begin{align} f_i = \text{max}\left(1 - \frac{c_{d,i}}{\text{min}\left(d_i/\Delta z_i, 1\right)}, 0\right), \end{align} \]
where $\Delta z_i$ is the layer thickness. This means that complete coverage $f_i = 0$ is already achieved at $c_{d,i} = \frac{d_i}{\Delta z_i}$. Assuming random overlap between the layers, one obtains
\[ \begin{align} \text{TCC} = 1 - \prod_{i=1}^{N_L}f_i.\tag{39.19}\label{eq:tcc_random_overlap} \end{align} \]
In reality, however, it can be assumed that the overlap between the layers is not random, but rather that clouds move e.g. B. extend vertically over several layers. This is expressed using a coherence length $l_c$ and generalizes Eq. (39.19) to
\[ \begin{align} \text{TCC} = 1 - f_1\prod_{i=2}^{N_L}f_i + \left(1 - f_i\right)\exp\left(-\frac{\left|z_i - z_{i-1}\right|}{l_{c,i}}\right).\tag{39.20}\label{eq:tcc_general_overlap} \end{align} \]
In the case of short coherence lengths $l_{c,i}\to 0$, Eq. (39.20) to Eq. (39.19) over, in the case of infinite coherence lengths $l_{c,i}\to\infty$ follows from (39.20) $\text{TCC} = 1 - f_1$, because then the clouds in all layers overlap perfectly. The cloud height $d_i$ and the coherence length $l_{c,i}$ now need to be determined. The following approach is chosen here:
\[ \begin{align} d_i &= 10\cdot L_{m,i},\\ l_{c,i} &= d_i, \end{align} \]
here $L_{m,i}$ is the vertical mixing path length.
Not all variables used in the model can be derived from theory ab initio, for example the diffusion coefficients used in the turbulence parameterizations. They usually do not have exactly the right value and are therefore also called semi-empirical coefficients. They should not be called constants because they are not. Of course, in order to make the model as accurate as possible, you should use the correct value for it. There are two methods available to determine this:
From the literature, this results in a reasonable interval within which one can freely choose the coefficient to be determined. Within this interval you simply choose it so that the model is as accurate as possible. Since one can tune multiple parameters in an atmospheric model, this becomes a multidimensional optimization problem that requires multiple iterations.
The following coefficients in the model can be tuned:
The remainder of this section discusses the tuning of the individual coefficients.