This chapter concerns the determination of quantities that are not direct model variables but are nonetheless expected of weather models by default.
The so-called 2-m-temperature is the air temperature 2 m above the earth's surface. According to the WMO standard, the near-surface temperature is measured there and must therefore also be output by models. To calculate it, one starts from the one-dimensional heat conduction equation Eq. (5.221)
\[ \begin{align} \frac{\partial\psi}{\partial t} = \kappa\frac{\partial^2\psi}{\partial z^2} \end{align} \]
with $\kappa > 0$ as the diffusion coefficient. This is now solved for $z>0$ subject to the boundary condition
\[ \begin{align} \psi\left(0,t\right) = \psi_0\exp\left(-i\omega t\right). \end{align} \]
One now makes the ansatz
\[ \begin{align} \psi\left(z,t\right) = A\exp\left(ikz - i\omega t\right), \end{align} \]
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 the negative sign can be excluded on physical grounds, since it leads to an exponentially growing amplitude. The solution is therefore
\[ \begin{align} \psi\left(z,t\right) &= \psi_0\exp\left(-i\omega t\right)\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right)\exp\left(i\sqrt{\frac{\omega}{2\kappa}}z\right)\nonumber\\ &= \psi\left(0,t\right)\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right)\exp\left(i\sqrt{\frac{\omega}{2\kappa}}z\right). \end{align} \]
This solution is now applied to the problem of determining the 2-m-temperature, i.e. $\psi \to T$. For $\omega$ one uses $\omega = \frac{2\pi}{24\:\text{h}}$, i.e. the angular frequency of the diurnal temperature cycle. To a first approximation, the phase component can be neglected. For $\kappa$ one inserts 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, without loss of generality, an offset $T_1$ has been added and $\psi_0$ has been renamed to $T_2$. In the atmosphere, the following boundary conditions apply here
\[ \begin{align} T\left(0\right) &= T_1 + T_2,\\ T\left(z_l\right) &= T_1 + T_2\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z\right), \end{align} \]
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 the ground. From this one obtains
\[ \begin{align} T_1 &= T_s - T_2,\\ T_l &= T_s - T_2 + T_2\exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right) = T_s - T_2\left[1 - \exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right)\right] \Rightarrow T_2 = \frac{T_s - T_l}{1 - \exp\left(-\sqrt{\frac{\omega}{2\kappa}}z_l\right)}. \end{align} \]
From this 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 by setting $z = 2\:\text{m}$.
By default, the wind is measured ten meters above the ground, where the vectors of a model layer are usually not located. For this, one uses the logarithmic wind profile Eq. (17.43), i.e.
\[ \begin{align} v_h\left(z_{10}\right) &= \frac{u_\star}{k}\ln\left(\frac{z_{10}}{z_0}\right),\\ v_h\left(z_l\right) &= \frac{u_\star}{k}\ln\left(\frac{z_l}{z_0}\right). \end{align} \]
Here $v_h\geq 0$ is the horizontal wind speed, $z_l$ is the height of the lowest model layer above the 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 holds only over water. Over land, the convention is that the wind must be measured over grass in order to ensure comparability. For this, one uses the equations
\[ \begin{align} v_h\left(z_{10}\right) &= \frac{u_\star}{k}\ln\left(\frac{z_{10}}{z_g}\right),\\ v_h\left(z_l\right) &= \frac{u_\star}{k}\ln\left(\frac{z_l}{z_0}\right), \end{align} \]
where $z_g$ is the roughness length of grass (0.02 m) and the friction velocity $u_\star$ is 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
\[ \begin{align} f_i = \text{max}\left(1 - \frac{c_{d,i}}{\text{min}\left(d_i/\Delta z_i, 1\right)}, 0\right) = 1 - c_i, \end{align} \]
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, one must assume that the overlap between the layers is not random, but rather that clouds extend vertically over several layers, for example. In the limiting case of maximum overlap, one has
\[ \begin{align} \text{TCC} = \text{max}\left(c_i\right).\tag{39.20}\label{eq:tcc_maximum_overlap} \end{align} \]
In reality, the total cloud cover usually lies between Eq. (39.19) and Eq. (39.20): clouds in adjacent layers are partly connected to one another and partly independent of one another. This is expressed through a coherence length $l_c$. An algorithmic procedure for obtaining a realistic total cloud cover is the following:
Compute $f_i$ in each layer $i$.
Set $F = f_1$. Here $F$ is the non-cloud-covered fraction of the layers considered so far.
Compute the cloud-covered part $\newtilde{c}_2$ of the fraction of the second layer that is not yet covered by clouds (whether the already cloud-covered fraction is covered by clouds or clear in the second layer is irrelevant for the total cloud cover). Here there are two limiting cases: in the case of random overlap, one has $\newtilde{c}_2 = 1 - f_2$. In the case of maximum overlap, one has $\newtilde{c}_2 = \text{max}\left(c_2 - C, 0\right)/F$, where $C = 1 - F$ is the cloud-covered fraction of the layers considered so far. In general, one has
\[ \begin{align} \newtilde{c}_2 &= w_2\cdot\frac{\text{max}\left(c_2 - C, 0\right)}{F} + \left(1 - w_2\right)c_2. \end{align} \]
Here the weight $w_2$ is computed via $w_2 = \exp\left(-\frac{\left|z_1 - z_2\right|}{l_{c,i}}\right)$.
Set $F \to F\cdot\left(1 - \newtilde{c}_2\right)$.
Repeat steps 3 to 4 for layers 3 to $N_L$.
Set $\text{TCC} = C$.
In the case of short coherence lengths $l_{c,i}\to 0$, Eq. (39.19) follows from this; in the case of infinite coherence lengths $l_{c,i}\to\infty$, Eq. (39.20) follows from this. 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.
Note that the algorithmic procedure presented here is not symmetric with respect to the direction of execution: it makes a difference whether one starts at the topmost layer and computes downwards, or starts at the bottommost layer and computes upwards. In reality, however, the cloudy fraction of the cross-sectional area of a grid column does not depend on whether one looks through it from below or from above. One could resolve this problem by computing both variants and then taking their mean. However, since both limiting cases Eq. (39.19) and Eq. (39.20) are symmetric, and the result of the presented algorithmic procedure lies between them, this problem is usually ignored.
Not all quantities used in the model can be derived ab initio from theory, for example the diffusion coefficients used in turbulence parameterizations. Their values are generally not known exactly and are therefore referred to as semi-empirical coefficients. They should not be called constants, because they are not. To make the model as accurate as possible, these coefficients should be chosen as realistically as possible. Two methods are available for estimating them:
They can be inferred from measurements.
They can be inferred from DNS runs (direct numerical simulation), i.e. very high-resolution simulations in the computational fluid dynamics (CFD) regime.
From the literature, this typically yields a plausible interval within which the coefficient can be selected. Within that interval, one chooses the value that maximizes model fidelity. Because multiple parameters can be tuned in an atmospheric model, this becomes a multidimensional optimization problem that requires multiple iterations.
The following model coefficients can be tuned:
horizontal diffusion coefficients
vertical diffusion coefficients
maximum specific cloud water
The remainder of this section discusses the tuning of these individual coefficients.