13 Important approximations

In meteorology and oceanography there are many difficult approximations with detailed and complex assumptions. This chapter explains the most common lists of them, starting with the less rigorous ones.

13.1 General

13.1.1 Filtering

Approximations are often made to the governing equations that are physically contradictory or contradict the definitions. Humidity definitely plays a role in the dynamics, but cyclogenesis (i.e. the formation of low pressure areas) can also be observed in a dry atmosphere. This is called filtering: Moisture is not necessary for the formation of low pressure areas because low pressure areas still form in a dry atmosphere and are therefore not filtered. If you want to investigate a phenomenon, you first always choose the simplest possible system of equations.

13.1.2 Dimensionless metrics

In order to be able to compare different terms in the governing equations, dimensionless key figures are introduced.

Nonlinearity is particularly problematic in the theoretical treatment of hydrodynamics. A large part of this lies in the momentum advection $\left(\mathbf{v}\cdot\nabla\right)\mathbf{v}$. These terms make a decisive contribution to the interaction of the scales (spectral components) and to the creation of instabilities. Friction, on the other hand, is a stabilizing influence. The so-called Reynolds number $N_\text{Re}$ is therefore defined as the ratio of these two terms:

\[ \begin{align} N_\text{Re} \coloneqq \frac{U^2}{L}\frac{L^2}{\nu U} = \frac{UL}{\nu} \end{align} \]

The larger the Reynolds number is, the more unstable and turbulent the flow is. On the synoptic scale applies

\[ \begin{align} N_\text{Re} \sim \frac{10^110^6}{10^{-5}} = 10^{12}. \end{align} \]

Synoptic currents are therefore very unstable.

The Rossby number $N_\text{Ro}$ is defined as the ratio of advective terms to the Coriolis force, i.e

\[ \begin{align} N_\text{Ro} \coloneqq \frac{U^2}{LfU} = \frac{U}{Lf}.\tag{13.3}\label{eq:def_rossby_number} \end{align} \]

13.2 Spherical geopotential approximation

Horizontal pressure gradients are four orders of magnitude smaller than vertical ones (see section 13.7), so it is practical to align one axis of the coordinate system with gravity so that it does not affect the horizontal equations of motion. First of all, you can simply use spherical coordinates for this; the orography can be interpreted as the height of the earth's surface above the sphere. However, the Earth is more of an ellipsoid than a sphere. A first approach to this could be to absorb the Earth's more ellipsoidal shape into the orography and continue to use spherical coordinates. The following applies to the angle $\varphi$ that the ellipsoid includes with a spherical surface

\[ \begin{align} \tan\left(\varphi\right) \approx \frac{\newtilde{f}}{\pi/2}\approx 0,2\:\%. \end{align} \]

Assuming that the gravity is perpendicular to the ellipsoid, it has a horizontal component in spherical coordinates of the order of magnitude $10^{-2}$ m/s$^2$, which is an order of magnitude larger than the horizontal pressure gradient. Overlaying the horizontal dynamics with such a large force is disadvantageous. So if you want to take the eccentricity of the gravity field into account, you should choose non-spherical coordinates.

However, if one calculates these effects into $g_z$ in terms of magnitude and uses a spherical approximation of the gravity field, one speaks of the spherical geopotential approximation (SGA). This means

\[ \begin{align} g_x &= 0,\\ g_y &= 0. \end{align} \]

Furthermore, one defines

\[ \begin{align} g \coloneqq -g_z. \end{align} \]

Thus, the equation system Equations (8.104) - (8.105) under the SGA becomes

$\mathbf{g}$ must be rotation-free as a gradient field,

\[ \begin{align} \nabla\times\mathbf{g} = \mathbf{0}. \end{align} \]

According to Eq. (B.114) applies

\[ \begin{align} \mathbf{0} &= \nabla\times\mathbf{g} = -\frac{g_y}{r}\mathbf{i} + \frac{g_x\tan\left(\varphi\right)}{r}\mathbf{k} + \frac{g_x}{r}\mathbf{j} + \mathbf{k}\left(-\frac{\partial g_x}{\partial y} + \frac{\partial g_y}{\partial x}\right) - \mathbf{j}\left(\frac{\partial g_z}{\partial x} - \frac{\partial g_x}{\partial r}\right) + \mathbf{i}\left(-\frac{\partial g_y}{\partial r} + \frac{\partial g_z}{\partial y}\right)\nonumber\\ & \stackrel{g_x = g_y = 0}{=} -\mathbf{j}\frac{\partial g_z}{\partial x} + \mathbf{i}\frac{\partial g_z}{\partial y}. \end{align} \]

Thus, in the SGA

\[ \begin{align} \frac{\partial g_z}{\partial x} = \frac{\partial g_z}{\partial y} = 0. \end{align} \]

So $g$ can only be a function of height,

The SGA is a relatively weak approximation, which greatly simplifies the system of equations. It is therefore almost always appropriate for analytical considerations. They can be given up for models, see section D.3.4.

13.3 Shallow atmosphere

The so-called shallow-atmosphere approximation is [32, 17]

for the functional determinant of the geographic coordinates, where $a$ is a constant value for the radius. Putting this into Eq. (B.98), you get

\[ \begin{align} \nabla\times\mathbf{v} &= \frac{1}{a^2\sin\left(\theta\right)}\Bigg[\mathbf{e}_r\left(\frac{\partial\left(a\sin\left(\theta\right)\newtilde{v}_\phi\right)}{\partial\theta} - \frac{\partial\left(a\newtilde{v}_\theta\right)}{\partial\phi}\right) + a\mathbf{e}_\theta\left(\frac{\partial\newtilde{v}_r}{\partial\phi} - \frac{\partial\left(a\sin\left(\theta\right)\newtilde{v}_\phi\right)}{\partial r}\right)\nonumber\\ & + a\sin\left(\theta\right)\mathbf{e}_\phi\left(\frac{\partial\left(a\newtilde{v}_\theta\right)}{\partial r} - \frac{\partial\newtilde{v}_r}{\partial\theta}\right)\Bigg]\nonumber\\ &= \frac{\newtilde{v}_\phi}{a\tan\left(\theta\right)}\mathbf{e}_r + \mathbf{e}_r\left(\frac{1}{a}\frac{\partial\newtilde{v}_\phi}{\partial\theta} - \frac{1}{a\sin\left(\theta\right)}\frac{\partial\newtilde{v}_\theta}{\partial\phi}\right)\nonumber\\ & + \mathbf{e}_\theta\left(\frac{1}{a\sin\left(\theta\right)}\frac{\partial\newtilde{v}_r}{\partial\phi} - \frac{\partial\newtilde{v}_\phi}{\partial r}\right) + \mathbf{e}_\phi\left(\frac{\partial \newtilde{v}_\theta}{\partial r} - \frac{1}{a}\frac{\partial\newtilde{v}_r}{\partial\theta}\right). \end{align} \]

This means that Eq. (B.115) in the shallow atmosphere the form

\[ \begin{align} \nabla\times\mathbf{v} &= \left(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}\right)\mathbf{i} + \left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}\right)\mathbf{j} + \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)\mathbf{k} + \frac{u\tan\left(\varphi\right)}{a}\mathbf{k}.\tag{13.17}\label{eq:rot_shallow} \end{align} \]

accepts. In the shallow atmosphere, the speed resulting from the earth's rotation applies

\[ \begin{align} \mathbf{v}_i &= \boldsymbol{\Omega}\times\mathbf{r} = \left(\begin{array}{c} 0\\ 0\\ \Omega \end{array}\right)\times a\left(\begin{array}{c} \cos\left(\varphi\right)\cos\left(\lambda\right)\\ \cos\left(\varphi\right)\sin\left(\lambda\right)\\ \sin\left(\varphi\right) \end{array}\right) = \Omega a\left(\begin{array}{c} -\cos\left(\varphi\right)\sin\left(\lambda\right)\\ \cos\left(\varphi\right)\cos\left(\lambda\right)\\ 0 \end{array}\right) = \Omega a\cos\left(\varphi\right)\left(\begin{array}{c} -\sin\left(\lambda\right)\\ \cos\left(\lambda\right)\\ 0 \end{array}\right)\nonumber\\ &= \Omega a\cos\left(\phi\right)\mathbf{i}. \end{align} \]

With Eq. (13.17) is obtained

\[ \begin{align} \nabla\times\mathbf{v}_i &= \left(-\frac{\partial\left(\Omega a\cos\left(\phi\right)\right)}{\partial y} + \Omega a\cos\left(\phi\right)\frac{\tan\left(\phi\right)}{a}\right)\mathbf{k}\nonumber\\ &= \left(-\frac{\partial\left(\Omega \cos\left(\phi\right)\right)}{\partial\phi} + \Omega\sin\left(\phi\right)\right)\mathbf{k} = 2\Omega\sin\left(\phi\right)\mathbf{k} \end{align} \]

Therefore, the shallow atmosphere approximation implies the so-called traditional approximation

Eq. (13.17) implies with Eq. (B.130) that in momentum advection the terms $\propto uw/r, vw/r, u^2/r, v^2/r$ must be neglected.

If you put Eq. (13.15) in Eq. (B.90), you get

\[ \begin{align} \nabla\cdot\mathbf{v} &= \frac{1}{a^2\sin\left(\theta\right)}\left(\frac{\partial\left(v^{(r)}a^2\sin\left(\theta\right)\right)}{\partial r} + \frac{\partial\left(v^{(\theta)}a^2\sin\left(\theta\right)\right)}{\partial\theta} + \frac{\partial\left(v^{(\phi)}a^2\sin\left(\theta\right)\right)}{\partial\phi}\right)\nonumber\\ &= \frac{\partial v^{(r)}}{\partial r} + \frac{\partial v^{(\theta)}}{\partial\theta} + \frac{\partial v^{(\phi)}}{\partial\phi} + \cot\left(\theta\right)v^{(\theta)}\nonumber\\ &= \frac{\partial\newtilde{v}^{(r)}}{\partial r} + \frac{1}{r}\frac{\partial\newtilde{v}^{(\theta)}}{\partial\theta} + \frac{1}{r\sin\left(\theta\right)}\frac{\partial\newtilde{v}^{(\phi)}}{\partial\phi} + \frac{\cot\left(\theta\right)}{a}\newtilde{v}^{(\theta)}. \end{align} \]

This means that Eq. (B.112) in the shallow atmosphere the form

\[ \begin{align} \nabla\cdot\mathbf{v} &= \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} - \frac{v\tan\left(\varphi\right)}{a} \end{align} \]

accepts. Since the density of the atmosphere is neglected when calculating $\mathbf{g}$, the following applies with the SGA

\[ \begin{align} \nabla\cdot\mathbf{g} = -\frac{dg}{dz} = 0, \end{align} \]

which implies that $g$ must be independent of height in the shallow atmosphere,

The implications of Eq. (13.15) are summarized as follows

1. in differential operators, $\frac{1}{r}$ must be replaced by $\frac{1}{a}$,
2. the traditional approximation $f' = 0$ must be done,
3. all metric terms that do not contain $\tan\left(\varphi\right)$ must be neglected (in divergence, vorticity and momentum advection),
4. the gravity must be independent of height, $g \to g_0 =$ homogeneous.

The momentum equation of the shallow atmosphere is therefore:

13.3.1 Modified Coriolis parameter

All approximations made so far are globally applicable. This is no longer the case with the approximation made in this section. One defines the modified Coriolis parameter $f^\star$ by

\[ \begin{align} f^\star \coloneqq f\left(1 + \frac{u}{2a\omega\cos\left(\varphi\right)}\right)\tag{13.28}\label{eq:f_mod}. \end{align} \]

Unlike $f$, $f^\star$ also depends on the velocity field and no longer just describes the Coriolis force. This means that the equations (13.25) - (13.26) can be written down shorter than

\[ \begin{align} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} &= -\frac{1}{\rho}\frac{\partial p}{\partial x} + f^\star v + F_{R,x} \tag{13.29}\label{eq:x_momentum_simplified_shallow_mod},\\ \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} &= -\frac{1}{\rho}\frac{\partial p}{\partial y} - f^\star u + F_{R,y} \tag{13.30}\label{eq:y_momentum_simplified_shallow_mod}. \end{align} \]

For most dynamic considerations one can assume $f^\star = f$, at least up to latitudes of 80 degrees. This is illustrated in Fig. 13.1. This gives the following simplified horizontal equations of motion:

13.4 Pseudo-incompressible approximation

The so-called pseudo-incompressible approximation was presented in [1]. In order to justify them, one first considers the equations of the shallow atmosphere in the form in Sect. 10.1.2

\[ \begin{align} \md{u} &= -c^{(p)}\theta\frac{\partial\Pi}{\partial x} + fv,\\ \md{v} &= -c^{(p)}\theta\frac{\partial\Pi}{\partial y} - fu,\\ \md{w} &= -c^{(p)}\theta\frac{\partial\Pi}{\partial z} - g,\\ \md{\rho} &= -\rho\nabla\cdot\mathbf{v},\\ \md{\Pi} & \stackrel{\href{ch-08-first-law-in-the-atmosphere.html#eq:exner-pressure_material_derivative}{\text{Glg. (9.79)}}}{=} -\frac{R_d\Pi}{c^{(V)}}\nabla\cdot\mathbf{v}.\tag{13.37}\label{eq:pseudo-inc_deriv_0} \end{align} \]

Only Eq. is now modified. (13.37). To do this, a background state $\left(\newoverline{\Pi}\left(z\right), \newoverline{\theta}\left(z\right)\right)$ is introduced; deviations from this are indicated as usual with primed values. This allows Eq. (13.37) assuming $\Pi' \ll \newoverline{\Pi}$ in the form

\[ \begin{align} \md{\Pi'} + w\frac{d\newoverline{\Pi}}{dz} + \frac{R_d\newoverline{\Pi}}{c^{(V)}}\nabla\cdot\mathbf{v} &= 0\\ \Leftrightarrow \frac{c^{(V)}}{R_d\Pi}\md{\Pi'} + \frac{c^{(V)}}{R_d\newoverline{\Pi}}w\frac{d\newoverline{\Pi}}{dz} + \nabla\cdot\mathbf{v} &= 0\tag{13.39}\label{eq:pseudo-inc_deriv_1} \end{align} \]

note down. The pseudo-incompressible approximation is now based on, in Eq. (13.39) to neglect the term $\frac{c^{(V)}}{R_d\Pi}\md{\Pi}$, i.e. from

\[ \begin{align} \frac{c^{(V)}}{R_d\newoverline{\Pi}}w\frac{d\newoverline{\Pi}}{dz} + \nabla\cdot\mathbf{v} = 0\tag{13.40}\label{eq:pseudo-inc_deriv_2} \end{align} \]

to go out. The background state satisfies the equation of state in the form Eq. (9.71):

\[ \begin{align} \newoverline{\Pi} &= \left(\frac{R_d\newoverline{\rho}\newoverline{\theta}}{p_0}\right)^{R_d/c^{(V)}} \end{align} \]

This follows from the chain rule

\[ \begin{align} \frac{d\newoverline{\Pi}}{dz} = \newoverline{\Pi}\frac{R_d}{c^{(V)}\newoverline{\rho}\newoverline{\theta}}\frac{d\left(\newoverline{\rho}\newoverline{\theta}\right)}{dz}. \end{align} \]

Putting this into Eq. (13.40), you get

\[ \begin{align} w\frac{d\left(\newoverline{\rho}\newoverline{\Pi}\right)}{dz} + \newoverline{\rho}\newoverline{\theta}\nabla\cdot\mathbf{v} = 0 \end{align} \]

This leads to the compact formulation

the pseudo-incompressible approximation.

13.5 Anelastic approximation

In the so-called anelastic approximation the thermodynamic quantities are written in the form

\[ \begin{align} \rho\left(\varphi, \lambda, z, t\right) &= \rho_0\left(z\right) + \rho'\left(\varphi, \lambda, z, t\right),\tag{13.45}\label{eq:anelastic_deriv_0}\\ p\left(\varphi, \lambda, z, t\right) &= p_0\left(z\right) + p'\left(\varphi, \lambda, z, t\right),\tag{13.46}\label{eq:anelastic_deriv_1}\\ \theta\left(\varphi, \lambda, z, t\right) &= \theta_0 + \theta'\left(\varphi, \lambda, z, t\right).\tag{13.47}\label{eq:anelastic_deriv_2} \end{align} \]

The background state $\left(\rho_0, p_0, \theta_0\right)$ is isentropic and hydrostatically balanced,

\[ \begin{align} \frac{dp_0}{dz} = -g\rho_0.\tag{13.48}\label{eq:anelastic_deriv_4} \end{align} \]

The unapproximated reversible equations are

\[ \begin{align} \rho\md{\mathbf{v}} &= -\nabla p - \rho\mathbf{f}\times\mathbf{v} + \rho\mathbf{g},\\ \frac{\partial\rho}{\partial t} + \nabla\cdot\left(\rho\mathbf{v}\right) &= 0,\\ \md{\theta} &= 0.\tag{13.51}\label{eq:anelastic_deriv_5} \end{align} \]

If you insert the equations (13.45) - (13.47) here. you get

\[ \begin{align} \left(\rho_0 + \rho'\right)\md{\mathbf{v}} &= -\nabla\left(p_0 + p'\right) - \left(\rho_0 + \rho'\right)\mathbf{f}\times\mathbf{v} + \left(\rho_0 + \rho'\right)\mathbf{g},\\ \frac{\partial\rho'}{\partial t} + \nabla\cdot\left[\left(\rho_0 + \rho'\right)\mathbf{v}\right] &= 0,\\ \md{\theta'} &= 0. \end{align} \]

The anelastic system of equations, like the incompressible one, is often used to study atmospheric deep convection.

13.5.1 Horizontal momentum equation

If you replace the density in the prefactor of the acceleration (including the Coriolis acceleration) by its mean value, you get

\[ \begin{align} \rho_0\md{\mathbf{v}} &= -\nabla\left(p_0 + p'\right) - \rho_0\mathbf{f}\times\mathbf{v} + \left(\rho_0 + \rho'\right)\mathbf{g}\nonumber\\ \Leftrightarrow \md{\mathbf{v}} &= -\frac{1}{\rho_0}\nabla\left(p_0 + p'\right) - \mathbf{f}\times\mathbf{v} + \frac{\rho_0 + \rho'}{\rho_0}\mathbf{g}\nonumber\\ \Leftrightarrow \md{\mathbf{v}} &= -\nabla\Phi - \mathbf{f}\times\mathbf{v} + \frac{\rho_0 + \rho'}{\rho_0}\mathbf{g}\tag{13.55}\label{eq:anelastic_deriv_3} \end{align} \]

with

\[ \begin{align} \Phi \coloneqq \frac{p'}{\rho_0} \end{align} \]

With the shallow atmosphere approximation one obtains the components of the horizontal momentum equation in the form

13.5.2 Vertical momentum equation

If you project Eq. (13.55) on $\mathbf{k}$, one obtains

\[ \begin{align} \md{w} &= -\frac{1}{\rho_0}\frac{\partial\left(p_0 + p'\right)}{\partial z} - \frac{\rho_0 + \rho'}{\rho_0}g\nonumber\\ \Leftrightarrow \md{w} &= -\frac{1}{\rho_0}\frac{\partial p_0}{\partial z} - g - \frac{1}{\rho_0}\frac{\partial p'}{\partial z} - \frac{\rho'}{\rho_0}g\nonumber\\ \Leftrightarrow \md{w} & \stackrel{\href{#eq:anelastic_deriv_4}{\text{Glg. (13.48)}}}{=} -\frac{1}{\rho_0}\frac{\partial p'}{\partial z} - \frac{\rho'}{\rho_0}g\nonumber\\ \Leftrightarrow \md{w} &= -\frac{\partial\Phi}{\partial z} - \frac{\Phi}{\rho_0}\frac{d\rho_0}{dz} - \frac{\rho'}{\rho_0}g\nonumber\\ \Leftrightarrow \md{w} &= -\frac{\partial\Phi}{\partial z} - \frac{p'}{\rho_0^2}\frac{d\rho_0}{dz} - \frac{\rho'}{\rho_0}g.\tag{13.59}\label{eq:anelastic_deriv_6} \end{align} \]

For the background temperature $T_0 = T_0\left(z\right)$ as a function of height applies

\[ \begin{align} T_0\left(z\right) = \theta_0\left(\frac{p_0}{p_\text{ref}}\right)^{R_d/c^{(p)}}. \end{align} \]

For the background density $\rho_0 = \rho_0\left(z\right)$ applies with the thermal equation of state of ideal gases

\[ \begin{align} \rho_0\left(z\right) = \frac{p_0}{R_dT_0} = \frac{p_0}{R_d\theta_0}\left(\frac{p_\text{ref}}{p_0}\right)^{R_d/c^{(p)}}.\tag{13.61}\label{eq:anelastic_deriv_7} \end{align} \]

From this it follows

\[ \begin{align} \frac{d\rho_0}{dz} = \rho_0\frac{1 - \frac{R_d}{c^{(p)}}}{p_0}\frac{dp_0}{dz} = -\rho_0\frac{1 - \frac{R_d}{c^{(p)}}}{p_0}g\rho_0 = -\frac{c^{(p)} - c^{(p)} + c^{(V)}}{c^{(p)}p_0}g\rho_0^2 = -\frac{c^{(V)}}{c^{(p)}p_0}g\rho_0^2 = -\frac{g\rho_0^2}{\kappa p_0}. \end{align} \]

Putting this into Eq. (13.59), you get

\[ \begin{align} \md{w} &= -\frac{\partial\Phi}{\partial z} + \frac{gp'}{\kappa p_0} - \frac{\rho'}{\rho_0}g = -\frac{\partial\Phi}{\partial z} + g\left(\frac{p'}{\kappa p_0} - \frac{\rho'}{\rho_0}\right). \end{align} \]

From Eq. (13.61) follows

\[ \begin{align} \theta\left(\rho, p\right) = \frac{p}{R_d\rho}\left(\frac{p_\text{ref}}{p}\right)^{R_d/c^{(p)}}. \end{align} \]

If you expand this to first order around $\left(\rho_0, p_0\right)$, you get

\[ \begin{align} \theta \approx \theta_0\left(1 - \frac{\rho'}{\rho_0} + \frac{p'}{\kappa p_0}\right) \Rightarrow \theta' \approx \theta_0\left(-\frac{\rho'}{\rho_0} + \frac{p'}{\kappa p_0}\right). \end{align} \]

This means that the approximately sign is ignored

\[ \begin{align} g\left(\frac{p'}{\kappa p_0} - \frac{\rho'}{\rho_0}\right) = g\frac{\theta'}{\theta_0}. \end{align} \]

Define the buoyancy $b$ by

\[ \begin{align} b \coloneqq g\frac{\theta'}{\theta_0}, \end{align} \]

this is the vertical momentum equation of the anelastic approximation

13.5.3 Temperature equation

The buoyancy $b$ is only a function of the potential temperature $\theta$, so it follows from Eq. (13.51)

13.5.4 Continuity equation

If you put Eq. (13.45) into the continuity equation, you get

\[ \begin{align} \frac{\partial \rho'}{\partial t} + \nabla\cdot\left[\left(\rho_0 + \rho'\right)\mathbf{v}\right] = 0. \end{align} \]

If you neglect the fluctuation $\rho'$, you get

13.5.5 Compilation

The anelastic system of equations is summarized as follows

13.6 Boussinesq approximation

In [2] the so-called Boussinesq approximation was derived for an ideal gas. However, nowadays it is mainly applied to the ocean, so the derivation in [2] is generalized here for a general fluid.

Let $\psi$ be one of the thermodynamic state variables. Note for these

\[ \begin{align} \psi = \psi\left(\varphi, \lambda, z\right) = \psi_m + \psi_0\left(z\right) + \psi'\left(\varphi, \lambda, z\right),\tag{13.77}\label{eq:boussinesq_deriv_5} \end{align} \]

where $\psi_m$ is the mean of $\psi$, $\psi_0$ is the hydrostatic stratification in the absence of motion and accelerations and $\psi'$ is the variation caused by motion. Define the scale height corresponding to $\psi$ $D_\psi$ by

\[ \begin{align} D_\psi \coloneqq \left|\frac{1}{\psi_m}\frac{d\psi_0}{dz}\right|^{-1}. \end{align} \]

The fluid has thickness $h$. The first part of the so-called Boussinesq approximation

where $\left(D_\psi\right)_\text{min}$ denotes the minimum scale height of all thermodynamic state variables. This implies

\[ \begin{align} \left|\frac{\psi_0}{\psi_m}\right| \ll 1\tag{13.80}\label{eq:boussinesq_deriv_0} \end{align} \]

for all thermodynamic variables. The second part of the Boussinesq approximation is

The Boussinesq approximation is justified in the ocean, but in the atmosphere only for shallow systems.

At this point we assume a homogeneous system. In this case one can write the thermal equation of state in the form

\[ \begin{align} \rho = \rho\left(p, T\right) \end{align} \]

note down. This can be developed into a Taylor series around the development point $\left(p, T\right)^T = \left(p_m, T_m\right)^T$:If you define $\rho_m, p_m, T_m$ as the mean of the respective quantities, due to the nonlinearity of the equation of state, in general $\rho_m \not= \rho\left(p_m, T_m\right)$ applies. Therefore, it makes sense to define $\rho_m \coloneqq \rho\left(p_m, T_m\right)$.

\[ \begin{align} \rho\left(p, T\right) &= \rho_m\big[1 - a_m\left(T - T_m\right) + K_m\left(p - p_m\right) + \frac{1}{2}\left(\frac{1}{\rho}\frac{\partial^2\rho}{\partial T^2}\right)_m\left(T - T_m\right)^2\nonumber\\ & + \frac{1}{2}\left(\frac{1}{\rho}\frac{\partial^2\rho}{\partial p^2}\right)_m\left(p - p_m\right)^2 + \left(\frac{1}{\rho}\frac{\partial^2\rho}{\partial T\partial p}\right)_m\left(T - T_m\right)\left(p - p_m\right)\nonumber\\ & + O\left[\left(T - T_m\right)^3, \left(p - p_m\right)^3\right]\big]\tag{13.83}\label{eq:taylor_eos_boussinesq} \end{align} \]

The definitions were

\[ \begin{align} a_m &\coloneqq -\left[\frac{1}{\rho}\left(\frac{\partial\rho}{\partial T}\right)_p\right]_m\text{(thermischer Expansionskoeffizient)},\\ K_m &\coloneqq \left[\frac{1}{\rho}\left(\frac{\partial\rho}{\partial p}\right)_T\right]_m\text{(Kompressibilität)} \end{align} \]

used. For ideal gases, $\rho = \frac{p}{R_sT}$

\[ \begin{align} a_m &= -\left[\frac{1}{\rho}\left(\frac{\partial\rho}{\partial T}\right)_p\right]_m = \frac{p_m}{R_s\rho T_m^2} = \frac{\rho_mR_sT_m}{R_s\rho_mT_m^2} = \frac{1}{T_m},\tag{13.86}\label{eq:therm_exp_id}\\ K_m &= \left[\frac{1}{\rho}\left(\frac{\partial\rho}{\partial p}\right)_T\right]_m = \frac{1}{\rho_m}\frac{1}{R_sT_m} = \frac{1}{p_m}.\tag{13.87}\label{eq:compress_id} \end{align} \]

For fluids you can

\[ \begin{align} a_m \ll -\frac{1}{\rho_m}\frac{-\rho_m}{T_m} = \frac{1}{T_m}, & {} & K_m \ll \frac{1}{\rho_m}\frac{\rho_m}{p_m} = \frac{1}{p_m}\tag{13.88}\label{eq:boussinesq_deriv_3} \end{align} \]

estimate. Putting this into Eq. (13.83) also for the second derivatives, which is obtained by neglecting the third and higher order terms

\[ \begin{align} \frac{\rho - \rho_m}{\rho_m} \ll -\frac{T - T_m}{T_m} + \frac{p - p_m}{p_m} \pm \frac{\left(T - T_m\right)^2}{T_m^2} \pm \frac{\left(p - p_m\right)^2}{p_m^2}. \end{align} \]

Based on equations (13.80) and (13.81) one can do the right side

\[ \begin{align} -\frac{T - T_m}{T_m} + \frac{p - p_m}{p_m} \pm \frac{\left(T - T_m\right)^2}{T_m^2} \pm \frac{\left(p - p_m\right)^2}{p_m^2} \approx -\frac{T - T_m}{T_m} + \frac{p - p_m}{p_m} \end{align} \]

approach. It therefore applies, neglecting the approximate sign (in the first order in the perturbations $\psi - \psi_m$ of the thermodynamic variable $\psi$)

\[ \begin{align} \frac{\rho - \rho_m}{\rho_m} = -a_m\left(T - T_m\right) + K_m\left(p - p_m\right).\tag{13.91}\label{eq:boussinesq_deriv_6} \end{align} \]

This implies

\[ \begin{align} \rho_0 &= \rho_m\left(-a_mT_0 + K_mp_0\right),\\ \rho' &= \rho_m\left(-a_mT' + K_mp'\right).\tag{13.93}\label{eq:boussinesq_deriv_2} \end{align} \]

13.6.1 Momentum equation

The basic hydrostatic equation (13.122) is

\[ \begin{align} \frac{dp_0}{dz} = -g\rho_m - g\rho_0.\tag{13.94}\label{eq:boussinesq_hydrostat} \end{align} \]

Plugging this into the momentum equation you get

\[ \begin{align} \rho\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)\mathbf{v} &= -\nabla p' - \frac{dp_0}{dz}\mathbf{k} - g\rho\mathbf{k} - \rho\mathbf{f}\times\mathbf{v} + \rho\mathbf{f}_R\nonumber\\ &= -\nabla p' - \frac{dp_0}{dz}\mathbf{k} - g\rho_m\mathbf{k} - g\rho_0\mathbf{k} - g\rho'\mathbf{k} - \rho\mathbf{f}\times\mathbf{v} + \rho\mathbf{f}_R\nonumber\\ &= -\nabla p' - g\rho'\mathbf{k} - \rho\mathbf{f}\times\mathbf{v} + \rho\mathbf{f}_R. \end{align} \]

Dividing this by $\rho_m$ gives

\[ \begin{align} \frac{\rho}{\rho_m}\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)\mathbf{v} &= -\frac{1}{\rho_m}\nabla p' - g\frac{\rho'}{\rho_m}\mathbf{k} - \frac{\rho}{\rho_m}\mathbf{f}\times\mathbf{v} + \frac{\rho}{\rho_m}\mathbf{f}_R. \end{align} \]

Due to Eq. (13.80) you can

\[ \begin{align} \frac{\rho}{\rho_m} \approx 1\tag{13.97}\label{eq:boussinesq_deriv_1} \end{align} \]

approach. This leads to

The continuity equation is:

\[ \begin{align} \left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)\rho = -\rho\nabla\cdot\mathbf{v}. \end{align} \]

Dividing this by $\rho_m$ gives

\[ \begin{align} \left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)\frac{\rho - \rho_m}{\rho_m} = -\frac{\rho}{\rho_m}\nabla\cdot\mathbf{v}. \end{align} \]

If you put Eq. (13.80), you get

So, under the Boussinesq approximation, the continuity equation simplifies to its incompressible form.

One can use the momentum equation Eq. (13.98) simplify a bit. The vertical component of this equation is

\[ \begin{align} \left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)w &= -\frac{1}{\rho_m}\frac{\partial p'}{\partial z} - g\frac{\rho'}{\rho_m} - \left(\mathbf{f}\times\mathbf{v}\right)\cdot\mathbf{k} + \mathbf{f}_R\cdot\mathbf{k}. \end{align} \]

For pressure gradient and gravity one obtains using Eq. (13.93)

\[ \begin{align} -\frac{1}{\rho_m}\frac{\partial p'}{\partial z} - g\frac{\rho'}{\rho_m} &= -\frac{1}{\rho_m}\frac{\partial p'}{\partial z} - g\left(-a_mT' + K_mp'\right) = -\frac{1}{\rho_m}\frac{\partial p' }{\partial z} - gK_mp' + ga_mT'\nonumber\\ &= -\frac{1}{\rho_m}\left(\frac{\partial p' }{\partial z} + g\rho_mK_mp'\right) + ga_mT' = -\frac{1}{\rho_m}\left(\frac{\partial p' }{\partial z} + \frac{p'}{H}\right) + ga_mT'\tag{13.103}\label{eq:boussinesq_deriv_4} \end{align} \]

with

\[ \begin{align} H \coloneqq \frac{1}{g\rho_mK_m}. \end{align} \]

With Eq. (13.88) you can

\[ \begin{align} H = \frac{1}{g\rho_m}\frac{1}{K_m} \gg \frac{1}{g\rho_m}p_m =: H' \end{align} \]

estimate. $H'$ is the thickness of a fluid of homogeneous density $\rho_m$, which rests hydrostatically in the gravity field $g$ and in which the pressure increases linearly from top to bottom from zero to $p_m$. One can

\[ \begin{align} H' \sim \frac{h}{2} \end{align} \]

estimate, so it also applies

\[ \begin{align} H \gg h. \end{align} \]

This means that in Eq. (13.103)

\[ \begin{align} \frac{\partial p' }{\partial z} + \frac{p'}{H} \approx \frac{\partial p'}{\partial z} \end{align} \]

can approach. This leads to a simplified form of Eq. (13.98):

In the ideal gas with Eq. (13.86)

\[ \begin{align} \left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)\mathbf{v} &= -\frac{1}{\rho_m}\nabla p' + g\frac{T'}{T_m}\mathbf{k} - \mathbf{f}\times\mathbf{v} + \mathbf{f}_R. \end{align} \]

13.6.2 Temperature equation

If you multiply Eq. (9.10) with density $\rho$ is obtained

\[ \begin{align} \rho c^{(V)}\md{T} - \frac{p}{\rho}\md{\rho} = q^{(V)},\tag{13.111}\label{eq:boussineq_t_deriv_0} \end{align} \]

where $q^{(V)}$ is the thermal power density. Due to Eq. (13.77) applies

\[ \begin{align} \md{T} = \frac{\partial T}{\partial t} + \mathbf{v}\cdot\nabla T = \frac{\partial T'}{\partial t} + \mathbf{v}\cdot\nabla T. \end{align} \]

Putting this into Eq. (13.111), you get

\[ \begin{align} \rho_mc^{(V)}\left(\frac{\partial T'}{\partial t} + \mathbf{v}\cdot\nabla T\right) - \frac{p}{\rho}\md{\rho} = q^{(V)},\tag{13.113}\label{eq:boussinesq_deriv_7} \end{align} \]

where in the prefactor of the temperature tendency the density $\rho$ was replaced by the mean density $\rho_m$. The compression term can be found in the first order in the deviations from the average values

\[ \begin{align} -\frac{p}{\rho}\md{\rho} \approx -\frac{p_m}{\rho_m}\md{\rho} = -\frac{p_m}{\rho_m}\left(\frac{\partial\rho}{\partial t} + \mathbf{v}\cdot\nabla\rho\right) \end{align} \]

simplify. With Eq. (13.91) follows, ignoring the approximately sign

\[ \begin{align} -\frac{p}{\rho}\md{\rho} \approx -\frac{p_m}{\rho_m}\left(\frac{\partial\rho}{\partial t} + \mathbf{v}\cdot\nabla\rho\right) = -p_m\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)\left[-a_m\left(T - T_m\right) + K_m\left(p - p_m\right)\right]. \end{align} \]

In order to eliminate the time dependence of the pressure, one approaches further

\[ \begin{align} -\frac{p}{\rho}\md{\rho} & \approx -p_m\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)\left[-a_m\left(T - T_m\right) + K_m\left(p - p_m\right)\right]\nonumber\\ &= -p_m\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)\left[-a_m\left(T_0 + T'\right) + K_mp_0\right]\nonumber\\ &= p_m\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)a_mT' + p_mwa_m\frac{dT_0}{dz} - p_mK_mw\frac{dp_0}{dz}. \end{align} \]

According to Eq. (13.94) applies to the first order in the deviation from the mean density

\[ \begin{align} \frac{dp_0}{dz} \approx -g\rho_m. \end{align} \]

This means that the approximately sign is ignored

\[ \begin{align} -\frac{p}{\rho}\md{\rho} &= p_m\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)a_mT' + p_mwa_m\frac{dT_0}{dz} - p_mK_mw\frac{dp_0}{dz}\nonumber\\ &= p_m\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)a_mT' + p_mwa_m\frac{dT_0}{dz} + p_mK_mwg\rho_m. \end{align} \]

Putting this into Eq. (13.113), you get

\[ \begin{align} & \rho_mc^{(V)}\left(\frac{\partial T'}{\partial t} + \mathbf{v}\cdot\nabla T\right) - \frac{p}{\rho}\md{\rho} = q^{(V)}\nonumber\\ & \rho_mc^{(V)}\left(\frac{\partial T'}{\partial t} + \mathbf{v}\cdot\nabla T\right) + p_m\left(\frac{\partial}{\partial t} + \mathbf{v}\cdot\nabla\right)a_mT' + p_mwa_m\frac{dT_0}{dz} + p_mK_mwg\rho_m = q^{(V)}\nonumber\\ & \left(\rho_mc^{(V)} + p_ma_m\right)\left(\frac{\partial T'}{\partial t} + \mathbf{v}\cdot\nabla T'\right) + w\left(p_ma_m + \rho_mc^{(V)}\right)\frac{dT_0}{dz} + p_mK_mwg\rho_m = q^{(V)} \end{align} \]

This is the temperature equation of the Boussinesq approximation:

For the ideal gas, the following equations apply (13.86) - (13.87)

\[ \begin{align} \left(c^{(V)} + R_s\right)\left(\frac{\partial T'}{\partial t} + \mathbf{v}\cdot\nabla T'\right) + w\left[\left(R_s + c^{(V)}\right)\frac{dT_0}{dz} + g\right] &= \frac{q^{(V)}}{\rho_m}\nonumber\\ \Leftrightarrow c^{(p)}\left(\frac{\partial T'}{\partial t} + \mathbf{v}\cdot\nabla T'\right) + wc^{(p)}\left(\frac{dT_0}{dz} + g\right) &= \frac{q^{(V)}}{\rho_m}\nonumber\\ \Leftrightarrow\left(\frac{\partial T'}{\partial t} + \mathbf{v}\cdot\nabla T'\right) + w\left(\frac{dT_0}{dz} + \frac{g}{c^{(p)}}\right) &= \frac{q^{(V)}}{c^{(p)}\rho_m}. \end{align} \]

The pseudo-incompressible, anelastic and Boussinesq approximations are summarized under the term soundproof because they filter sound waves.

13.7 Hydrostatics

The pressure gradient and gravity are of the order of $10$ m/s$^2$, while the vertical acceleration is of the order of $10^{-7}$ m/$s^2$. The equation

is therefore valid on the synoptic scale in an excellent approximation, this is the hydrostatic basic equation.

Another formulation of this equation is $\md{w} = 0$. The particles therefore have a constant vertical speed. This must be zero, otherwise it will be the rule that a particle disappears into the ground or space. So the implication applies

\[ \begin{align} \frac{\mathbf{g}}{g}\cdot\nabla p = \rho\mathbf{g} \Rightarrow w = 0. \end{align} \]

Nevertheless, it makes sense to allow a vertical velocity in hydrostatic equation systems. Although this is physically contradictory, it is not mathematically contradictory, since the system of equations cannot know anything about the above conclusion. The vertical movements of such a system arise from the continuity equation.

The basic hydrostatic equation makes it possible to use the pressure p not as a dependent but as an independent coordinate (vertical coordinate). If you transform from $p\left(\Phi\right)$ to $\Phi\left(p\right)$, this is the basic hydrostatic equation

with $\alpha \coloneqq \frac{1}{\rho}$ as the specific volume. The equation of state follows $\alpha = \frac{R_dT}{p}$ in a dry atmosphere. That's why $\frac{\partial\Phi}{\partial p}$ is often simply referred to as „ temperature“. It can be seen that the layer thickness is proportional to the temperature of the layer. The three-dimensional geopotential field is an expression of the ground pressure and the temperature of the air masses.

13.7.1 Continuity equation

The continuity equation is general

\[ \begin{align} \frac{\partial\rho}{\partial t} + \nabla\cdot\left(\rho\mathbf{v}\right) = 0. \end{align} \]

If one inserts the basic hydrostatic equation $\rho = -\frac{1}{g}\frac{\partial p}{\partial z}$, it follows

\[ \begin{align} \frac{\partial^2p}{\partial z\partial t} + \nabla\cdot\left(\frac{\partial p}{\partial z}\mathbf{v}\right) &= 0\nonumber\\ \Leftrightarrow\frac{\partial^2p}{\partial z\partial t} + \mathbf{v}\cdot\nabla\left(\frac{\partial p}{\partial z}\right) + \frac{\partial p}{\partial z}\nabla\cdot\mathbf{v} &= \frac{\partial^2p}{\partial z\partial t} + \frac{\partial}{\partial z}\left(\mathbf{v}\cdot\nabla p\right) - \nabla p\cdot\frac{\partial\mathbf{v}}{\partial z} + \frac{\partial p}{\partial z}\nabla\cdot\mathbf{v}\nonumber\\ = \frac{\partial\omega}{\partial z} - \nabla_hp\cdot\frac{\partial\mathbf{v}_h}{\partial z} + \frac{\partial p}{\partial z}\nabla\cdot\mathbf{v}_h &= 0. \end{align} \]

A small transformation term was neglected, see Eq. (B.112). According to the chain rule, $\frac{\partial}{\partial z} = \frac{\partial p}{\partial z}\frac{\partial }{\partial p}$. This follows

\[ \begin{align} \frac{\partial\omega}{\partial p} + \nabla\cdot\mathbf{v}_h - \nabla_hp\cdot\frac{\partial\mathbf{v}_h}{\partial p} = 0. \end{align} \]

If you use Eq. (12.7), you get

\[ \begin{align} \left(\frac{\partial u}{\partial x}\right)_z &= \left(\frac{\partial u}{\partial x}\right)_p + \frac{\partial u}{\partial p}\left(\frac{\partial p}{\partial x}\right)_z\Leftrightarrow\left(\frac{\partial u}{\partial x}\right)_z - \frac{\partial u}{\partial p}\left(\frac{\partial p}{\partial x}\right)_z = \left(\frac{\partial u}{\partial x}\right)_p \end{align} \]

and analogously for $v$. Unspecified partial derivatives have so far been derivatives in the z system. Now derivatives are used in the p-system:

In the p-system, the continuity equation is reduced to a purely diagnostic equation, which formally corresponds to the continuity equation for an incompressible fluid in the z-system. If you scale this equation with the values ​​from tab. 1.2, it follows in SI units

• $\nabla\cdot\mathbf{v}_h\sim10^{-5}$
• $\frac{\partial\omega}{\partial p}\sim10^{-6}$.

So the divergence of the horizontal wind is actually an order of magnitude smaller than the synoptic-scale vorticity.

13.7.2 Horizontal momentum equation

If you want to transform the equations of motion for flat geofluids (13.31) - (13.32), only the total derivatives in the p-system have to be noted on the left side. The pressure gradient takes a little more work. With Eq. (12.7) and the hydrostatic approximation follows

\[ \begin{align} \left(\frac{\partial p}{\partial x}\right)_p = 0 = \left(\frac{\partial p}{\partial x}\right)_z + \left(\frac{\partial z}{\partial x}\right)_p\frac{\partial p}{\partial z} = \left(\frac{\partial p}{\partial x}\right)_z - g\rho\left(\frac{\partial z}{\partial x}\right)_p. \end{align} \]

So it applies

\[ \begin{align} \frac{1}{\rho}\left(\frac{\partial p}{\partial x}\right)_z = g\left(\frac{\partial z}{\partial x}\right)_p = \left(\frac{\partial\Phi}{\partial x}\right)_p \end{align} \]

and analogously in the y direction. This means that the horizontal momentum equations in the p-system become

13.7.3 Temperature equation

If you derive the equation of state $p\alpha = R_dT$ in time, it follows

\[ \begin{align} \md{\left(p\alpha\right)} &= \md{\left(R_dT\right)}\nonumber\\ \Rightarrow\omega\alpha + p\md{\alpha} &= R_d\md{T}. \end{align} \]

From Eq. (9.11) follows ignoring the mass source density

\[ \begin{align} p\md{\alpha} = -c^{(V)}\md{T} + \alpha q_T^{(V)}. \end{align} \]

Thus you get

\[ \begin{align} \omega\alpha - c^{(V)}\md{T} + \alpha q_T^{(V)} = R_d\md{T}\Leftrightarrow - \omega\alpha + \left(R_d + c^{(V)}\right)\md{T} &= c^{(p)}\md{T} - \alpha\omega = \alpha q_T^{(V)}\nonumber\\ \md{_hT} - \omega\left(\frac{\alpha}{c^{(p)}} - \frac{\partial T}{\partial p}\right) &= \frac{\alpha}{c^{(p)}}q_T^{(V)}. \end{align} \]

With the equation of state follows

this defines the stability parameter $S_p$. You receive

This should now be rewritten as an equation for the evolution of $\Phi = \Phi\left(\phi, \lambda, p, t\right)$. From Eq. (13.124) follows

\[ \begin{align} \frac{\partial\Phi}{\partial p} = -\frac{R_dT}{p}\Rightarrow T = -\frac{p}{R_d}\frac{\partial\Phi}{\partial p}. \end{align} \]

This is now put into Eq. (13.138), it follows

\[ \begin{align} \md{_h}\left(-\frac{p}{R_d}\frac{\partial\Phi}{\partial p}\right) - S_p\omega &= \frac{\alpha}{c^{(p)}}q_T^{(V)}\nonumber\\ \Rightarrow\md{_h}\left(\frac{p}{R_d}\frac{\partial\Phi}{\partial p}\right) + S_p\omega &= -\frac{\alpha}{c^{(p)}}q_T^{(V)}\nonumber\\ \Rightarrow\md{_h}\left(\frac{\partial\Phi}{\partial p}\right) + \frac{R_dS_p}{p}\omega &= -\frac{R_d\alpha}{pc^{(p)}}q_T^{(V)}\nonumber\\ \Rightarrow\left(\frac{\partial}{\partial t} + u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y}\right)\left(\frac{\partial\Phi}{\partial p}\right) + \frac{R_dS_p}{p}\omega &= -\frac{R_d\alpha}{pc^{(p)}}q_T^{(V)}. \end{align} \]

The stability parameter $S_p$ is related to the vertical gradient of potential temperature:

\[ \begin{align} S_p = \frac{R_dT}{c^{(p)}p} - \frac{\partial T}{\partial p} = -\frac{T}{\theta}\left(\frac{\partial T}{\partial p} - T\frac{1}{p}\frac{R_d}{c^{(p)}}\right)\left(\frac{p_0}{p}\right)^{\frac{R_d}{c^{(p)}}} = -\frac{T}{\theta}\frac{\partial}{\partial p}\left[T\left(\frac{p_0}{p}\right)^{\frac{R_d}{c^{(p)}}}\right] = -\frac{T}{\theta}\frac{\partial\theta}{\partial p}\tag{13.141}\label{eq:stabilitaetspara_vereinfacht} \end{align} \]

Now define the static stability parameter $\sigma$ by

\[ \begin{align} \sigma \coloneqq\frac{R_dS_p}{p}\stackrel{\text{Glg. }\href{#eq:stabilitaetspara_vereinfacht}{(13.141)}}{=} - \frac{R_d}{p}\frac{T}{\theta}\frac{\partial\theta}{\partial p} = -\frac{\alpha}{\theta}\frac{\partial\theta}{\partial p}. \end{align} \]

This depends on Eq. (13.122) about

\[ \begin{align} \sigma = -\frac{\alpha}{\theta}\frac{\partial z}{\partial p}\frac{\partial\theta}{\partial z} = \frac{\alpha^2}{g\theta}\frac{\partial\theta}{\partial z} = \frac{\alpha^2}{g^2}\frac{g}{\theta}\frac{\partial\theta}{\partial z} = \left(\frac{\alpha}{g}N\right)^2 \end{align} \]

with the Brunt-Väisälä frequency $N$. You receive

13.7.4 Summary

The state of a dry hydrostatic atmosphere is determined by the fields $\left(u, v, \Phi\right)^T$, where all fields depend on $\left(\phi, \lambda, p, t\right)^T$. The governing equations are as follows:

As you can see, $\omega$ is a diagnostic quantity. $\alpha$ can be calculated from $\Phi$ using Eq. (13.124) can be determined, then the temperature can also be calculated diagnostically via $T = \frac{p\alpha}{R_d}$.

13.7.5 Hydrostatic background condition

Even if one does not make the hydrostatic assumption, one can introduce a ground state $\{\newoverline{\alpha}, \newoverline{p}\}$ with $\alpha \coloneqq \frac{1}{\rho}$ as the specific volume, the Eq. (13.122) is fulfilled, and the actual quantities $\{\alpha, p\}$ are to be noted as an overlay of the ground state with deviations $\{\alpha', p'\}$, i.e

\[ \begin{align} \alpha &= \newoverline{\alpha} + \alpha',\\ p &= \newoverline{p} + p'. \end{align} \]

Then applies

\[ \begin{align} -\alpha\nabla p + \mathbf{g} = -\alpha'\nabla\newoverline{p} - \newoverline{\alpha}\nabla p' - \alpha'\nabla p' = -\alpha'\nabla p - \newoverline{\alpha}\nabla p' = -\alpha\nabla p' - \alpha'\nabla\newoverline{p}.\tag{13.151}\label{eq:background_state_prop_0} \end{align} \]

The Earth's gravity field has a complicated shape that can be approached in different stages. First of all, it is assumed that the gravity field is radially symmetric with a homogeneous magnitude of the gravity vector. If you use the geopotential height $\Phi = gz$ instead of the geometric height $z$ (this is the energy per mass that is necessary to bring a particle from sea level to the height $z$), the basic hydrostatic equation with the chain rule is

\[ \begin{align} \frac{\partial p}{\partial\Phi} = \frac{\partial p}{\partial z}\frac{\partial z}{\partial\Phi} = -\rho. \end{align} \]

The SI unit of geopotential is m$^2$/s$^2$. A geopotential meter gpm is defined by

\[ \begin{align} 1\:\text{gpm} \coloneqq 9,8\:\text{m}^2/\text{s}^2, \end{align} \]

so that

\[ \begin{align} \frac{z}{\text{m}} = \frac{\phi/g}{\text{m}} \approx \frac{\phi}{\text{gpm}} \end{align} \]

applies. If you want to take into account the height dependence of gravitational acceleration, you first use the formula for the gravitational potential

\[ \begin{align} \Phi_g\left(r\right) = -\frac{\Phi_0a}{r} + \Phi_0 \end{align} \]

with $\Phi_0 \coloneqq GM/a$ with $G$ as Newton's gravitational constant and $M$ as Earth's mass. This is the formula for the gravity field of a planet with a radially symmetric mass distribution; it results from the formula for a point mass using Gauss's theorem. At $r = a$ the potential is normalized to zero. From this it follows

\[ \begin{align} g = \frac{\Phi_0a}{r^2} = g_0\frac{a^2}{r^2} \end{align} \]

with $g_0 \coloneqq \frac{\Phi_0}{a}$. In this approximation one requires that the generalized vertical coordinate geopotential height $z_g$ holds

\[ \begin{align} g_0z_g\hastobe\Phi_g\left(a + z\right) = -\frac{\Phi_0a}{a + z} + \Phi_0 \Rightarrow z_g = -\frac{a^2}{a + z} + a = \frac{-a^2 + a^2 + za}{a + z} = \frac{z}{1 + \frac{z}{a}}. \end{align} \]

It still follows from this

\[ \begin{align} \frac{\partial p}{\partial z} &= \frac{z_g}{z}\frac{\partial p}{\partial z_g} = \left(\frac{1}{1 + \frac{z}{a}} - \frac{z}{a}\frac{1}{\left(1 + \frac{z}{a}\right)^2}\right)\frac{\partial p}{\partial z_g} = \frac{1}{\left(1 + \frac{z}{a}\right)^2}\frac{\partial p}{\partial z_g} = \frac{a^2}{r^2}\frac{\partial p}{\partial z_g} = -g\rho = -g_0\frac{a^2}{r^2}\rho\nonumber\\ \Rightarrow \frac{\partial p}{\partial z_g} &= -g_0\rho. \end{align} \]

13.8 Barotropy

Barotropy is when the contour areas of the pressure field and the density field are the same. In a dry atmosphere this is equivalent to the fact that the temperature surfaces and the pressure surfaces are equal. In reality, the baroclinity angle, which describes the angle between $\nabla\rho$ and $\nabla p$, is small. However, the existing baroclinicity is very important for atmospheric dynamics and barotropy has far-reaching limiting implications.

Deriving the x component of Eq. (13.132) assuming barotropy to $p$, one obtains

\[ \begin{align} \frac{\partial}{\partial p}\md{u} = -\frac{\partial }{\partial p}\frac{\partial\Phi}{\partial x} + f\frac{\partial v}{\partial p} = -\frac{\partial }{\partial x}\frac{\partial\Phi}{\partial p} + f\frac{\partial v}{\partial p} = f\frac{\partial v}{\partial p}. \end{align} \]

It also applies

\[ \begin{align} \frac{\partial }{\partial p}\md{u} = \frac{\partial}{\partial p}\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + \omega\frac{\partial u}{\partial p}\right), \end{align} \]

it follows from this

\[ \begin{align} \frac{\partial}{\partial t}\frac{\partial u}{\partial p} = \frac{\partial}{\partial p}\frac{\partial u}{\partial t} = f\frac{\partial v}{\partial p} - \frac{\partial}{\partial p}\left(u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + \omega\frac{\partial u}{\partial p}\right). \end{align} \]

From this equation you can see that if the vertical shear disappears globally for one point in time, it will do so for all points in time. If the vertical shear disappears, the horizontal divergence and thus according to Eq. (13.129) $\frac{\partial\omega}{\partial p}$ constant height. The vertical velocity field in the p-system $\omega = \omega\left( p\right)$ is in this case a straight line and a diagnostic quantity.

13.8.1 Shallow water equations

The simplest system of equations in geofluid dynamics with time derivatives is that of shallow water equations (SWEs). To derive this, one starts from the equations (13.31) - (13.32) and assumes a homogeneous density and also ignores the friction. This makes the continuity equation:

\[ \begin{align} \nabla\cdot\mathbf{v} = 0. \end{align} \]

The fluid still has a surface where $p = 0$. Let the depth of the medium be $h = h\left(x, y, t\right)$ and the ground have the z coordinate $b = b\left(x, y\right)$. This is obtained by integrating the basic hydrostatic equation

\[ \begin{align} p\left(z\right) = -\left(0 - p\left(z\right)\right) = -\left(p\left(b + h\right) - p\left(z\right)\right) = -\int_{z}^{b + h}\frac{\partial p}{\partial z}dz = g\rho\int_{z}^{b + h}dz = \left(b + h - z\right)g\rho. \end{align} \]

Thus follows

\[ \begin{align} \frac{\partial p}{\partial x} = g\rho \frac{\partial}{\partial x}\left(b + h\right) \end{align} \]

and analogously in the y direction. If the vertical shear of the horizontal movement $\left(\frac{\partial u}{\partial z}, \frac{\partial v}{\partial z}\right)^T$ disappears globally at any point in time, the local time tendency of the vertical shear also disappears. The velocity field is therefore independent of height at all times. This is now assumed. This means that the terms of vertical velocity advection disappear. Now one can trivially integrate the continuity equation:

\[ \begin{align} h\nabla\cdot\mathbf{v} = \int_{b}^{b + h}\nabla\cdot\mathbf{v}dz = -\int_{b}^{b + h}\frac{\partial w}{\partial z}dz = -w\left(b + h\right) + w\left(b\right)\tag{13.165}\label{eq:swe_deriv_1} \end{align} \]

The assumption that the horizontal velocity is not sheared and that the integration Eq. (13.165) gives the shallow water equations their name. At great depths, this can clearly no longer be assumed. In Section 16.5.2 we go beyond this assumption. When dealing with waves, one can use the shallow water equations if

\[ \begin{align} \text{Wellenlänge} &\gg \text{Tiefe},\\ \text{Wellenhöhe} &\ll \text{Tiefe} \end{align} \]

apply, which can be the case for the tide (except in marginal seas) and the swell on the open sea.

At depth $b$ the kinematic boundary condition applies:

\[ \begin{align} w\left(b\right) &= \frac{dz}{dt}\left(b\right) = \frac{db}{dt} = \mathbf{v}\cdot\nabla_hb \end{align} \]

On the surface applies

\[ \begin{align} w\left(b + h\right) = \md{}\left(b + h\right) = \frac{\partial h}{\partial t} + \mathbf{v}_h\cdot\nabla_h\left(b + h\right). \end{align} \]

Putting this into Eq. (13.165), you get

\[ \begin{align} h\nabla_h\cdot\mathbf{v} &= -\frac{\partial h}{\partial t} - \mathbf{v}\cdot\nabla_h h. \end{align} \]

This results in the following system of equations:

13.8.1.1 Linearization

Assuming a homogeneous subsurface $b$ and an average depth $D$ on which a fault $d$ is superimposed, and neglecting all nonlinear terms, it follows

\[ \begin{align} \frac{\partial\mathbf{v}}{\partial t} &= -g\nabla d - f\mathbf{k}\times\mathbf{v},\tag{13.173}\label{eq:flach_lin_1}\\ \frac{\partial d}{\partial t} + D\nabla\cdot\mathbf{v} &= 0.\tag{13.174}\label{eq:flach_lin_2} \end{align} \]

13.9 Developments of the Coriolis parameter in plane geometry

For the Coriolis parameter $f$ as a function of the width $\varphi$ holds

\[ \begin{align} f\left(\phi\right) = 2\omega\sin\left(\phi\right). \end{align} \]

If you Taylor-expand this at a width $\phi_0$, you get

\[ \begin{align} f\left(\phi\right) &= f\left(\phi_0\right) + f'\left(\phi_0\right)\left(\phi - \phi_0\right) + \frac{1}{2}f''\left(\phi_0\right)\left(\phi - \phi_0\right)^2 + \frac{1}{6}f'''\left(\phi_0\right)\left(\phi - \phi_0\right)^2 + \mathcal{O}\left[\left(\phi - \phi_0\right)^4\right]\nonumber\\ &= f\left(\phi_0\right)\left[1 + \cot\left(\phi_0\right)\left(\phi - \phi_0\right) - \frac{1}{2}\left(\phi - \phi_0\right)^2 - \frac{1}{6}\cot\left(\phi_0\right)\left(\phi - \phi_0\right)^3\right], \end{align} \]

whereby the fourth and higher order terms were no longer noted. With

\[ \begin{align} y \coloneqq r\left(\phi - \phi_0\right) \end{align} \]

with $r$ as the distance from the center of the earth this can be written as

\[ \begin{align} f\left(\phi\right) &= f\left(\phi_0\right)\left[1 + \cot\left(\phi_0\right)\frac{y}{r} - \frac{y^2}{2r^2} - \frac{y^3}{6r^3}\cot\left(\phi_0\right)\right]\tag{13.178}\label{eq:f_dev} \end{align} \]

note down. The Rossby parameter is defined by

\[ \begin{align} \beta\left(\phi_0\right) \coloneqq \cot\left(\phi_0\right)\frac{f\left(\phi_0\right)}{r} = \frac{2\omega\cos\left(\phi_0\right)}{r} = \frac{d\phi}{dy}\frac{df}{d\phi} = \frac{df}{dy}. \end{align} \]

The approximations to be derived now assume a zonal channel of meridional extent $2B$, which is centered at the width $\varphi_0$. The curvature terms are neglected, which is why we speak of planes.

13.9.1 f-plane

This is where the approximation is made

\[ \begin{align} f\left(\phi\right) \approx f\left(\phi_0\right). \end{align} \]

According to Eq. (13.178) is this justified if

\[ \begin{align} B \ll r\tan\left(\phi_0\right), & {} & B \ll \sqrt{2}r \end{align} \]

apply. The so-called f-sphere assumes a globally homogeneous Coriolis parameter, which is not suitable for operational predictions.

13.9.2 $\beta-$plane

This is where the approximation is set

\[ \begin{align} f\left(\phi\right) \approx f\left(\phi_0\right) + \beta y \end{align} \]

to. According to Eq. (13.178) is this justified if

\[ \begin{align} B \ll 2r\cot\left(\phi_0\right), & {} & B \ll \sqrt{6}r \end{align} \]

apply.

13.10 Equilibrium winds

13.10.1 Geostrophy

Scaling the horizontal momentum equation on the synoptic scale.

Term	Magnitude in SI
$\frac{\partial u}{\partial t}$	$10^{-4}$
$u\frac{\partial u}{\partial x}$	$10^{-4}$
$fv$	$10^{-3}$
$\frac{1}{\rho}\frac{\partial p}{\partial x}$	$10^{-3}$

For the Rossby number $N_\text{Ro}$ applies

\[ \begin{align} N_\text{Ro} = \frac{U}{Lf} \sim10^{-1} \end{align} \]

on the synoptic scale. In the first order, there is a balance of forces consisting of the Coriolis force and the pressure gradient force. This is the geostrophic approximation. The corresponding wind field $\mathbf{v}_h = \left(u_g, v_g\right)^T$ is obtained by setting the acceleration equal to zero in the horizontal equations of motion (13.132) - (13.133).

\[ \begin{align} 0 = -\frac{\partial\Phi}{\partial x} + fv_g, & {} & 0 = -\frac{\partial\Phi}{\partial y} - fu_g \end{align} \]

If you adjust this according to the wind speed, you get:

\[ \begin{align} v_g = \frac{1}{f}\frac{\partial\Phi}{\partial x}, & {} & u_g = -\frac{1}{f}\frac{\partial\Phi}{\partial y}. \end{align} \]

So the geostrophic wind is

The geostrophic wind speed decreases as the latitude increases; at the equator it approaches infinity. It must therefore be questioned up to which latitudes geostrophy is a good approximation. This question will be examined in more detail in Section 13.9.2.

If you form the dot product with the gradient of the geopotential, it follows

\[ \begin{align} \nabla\Phi\cdot\mathbf{v}_{h, g} = 0, \end{align} \]

the geostrophic wind is perpendicular to the geopotential gradient. Therefore, the geostrophic wind is isohypse-parallel. In geostrophy, the local temporal tendency of the horizontal wind arises purely from advection. Further important implications of the geostrophic assumption are discussed in Sect. 13.9.

If one forms the divergence of Eq. (13.187), you get

\[ \begin{align} \nabla\cdot\mathbf{v}_{h, g} = -\frac{1}{f}\frac{\partial^2\Phi}{\partial x\partial y} + \frac{1}{f}\frac{\partial^2\Phi}{\partial y\partial x} - \frac{\beta}{f^2}\frac{\partial\Phi}{\partial x} - \frac{v\tan\left(\phi\right)}{r} = -\frac{\beta}{f^2}\frac{\partial\Phi}{\partial x} = -\frac{\beta}{f}v - \frac{v\tan\left(\phi\right)}{r}. \end{align} \]

Using the scales from tab. 1.3 follows

\[ \begin{align} \mathcal{O}\left(\nabla\cdot\mathbf{v}_{h, g}\right) = \frac{10^{-11}}{10^{-4}}10^{1}\:\frac{1}{\text{s}} = 10^{-6}\:\frac{1}{\text{s}} \end{align} \]

for medium and high latitudes. This is about an order of magnitude smaller than the synoptic-scale divergence, so the geostrophic wind can be viewed as almost divergence-free outside the tropics, but this no longer applies at lower latitudes. The geostrophic approximation is therefore not applicable globally, especially for models.

The thermal wind refers to the change of the geostrophic wind with height due to baroclinity. From Eq. (13.187) follows

The vertical shear of the geostrophic wind results from the baroclinicity of the atmosphere. One also speaks of vertical shear of the geostrophic wind due to horizontal temperature gradients, where horizontal here refers to the pressure area. Eq. (13.191) is the thermal wind equation.

Let $p_1 < p_2$ be two pressure levels, then by integrating Eq. (13.191)

\[ \begin{align} \mathbf{v}_{h, g}\left(p_2\right) - \mathbf{v}_{h, g}\left(p_1\right) &= \int_{p_1}^{p_2}-\frac{R_d}{pf}\mathbf{k}\times\nabla Tdp\nonumber\\ \Rightarrow\mathbf{v}_{h, g}\left(p_1\right) &= \mathbf{v}_{h, g}\left(p_2\right) + \int_{p_1}^{p_2}\frac{R_d}{pf}\mathbf{k}\times\nabla Tdp = \mathbf{v}_{h, g}\left(p_2\right) + \frac{R_d}{f}\int_{p_1}^{p_2}\frac{1}{p}\mathbf{k}\times\nabla Tdp\nonumber\\ &= \mathbf{v}_{h, g}\left(p_2\right) + \frac{R_d}{f}\ln\left(\frac{p_2}{p_1}\right)\mathbf{k}\times\nabla\newoverline{T}\left(p_1, p_2\right) \end{align} \]

with a weighted layer average temperature

\[ \begin{align} \newoverline{T}\left(p_1, p_2\right) &= \frac{\int_{p_1}^{p_2}\frac{1}{p}Tdp}{\int_{p_1}^{p_2}\frac{1}{p}dp}. \end{align} \]

13.10.2 Gradient wind

With the gradient wind it is assumed that the Coriolis force and the pressure gradient together produce the centripetal force that is necessary to move a fluid particle along a trajectory with radius $R > 0$, i.e

\[ \begin{align} \frac{V^2}{R} = \left|\left|f\right|V - \frac{1}{\rho}\left|\nabla_hp\right|\right|. \end{align} \]

In the cyclonic case, the expression between the outer magnitude signs is negative, i.e

\[ \begin{align} \frac{V^2}{R} &= \frac{1}{\rho}\left|\nabla_hp\right| - \left|f\right|V\nonumber\\ \Leftrightarrow V^2 + \left|f\right|VR - R\frac{1}{\rho}\left|\nabla_hp\right| &= 0 \nonumber\\ \Leftrightarrow\frac{V^2}{R} &= -\frac{1}{\rho}\left|\nabla_hp\right| + \left|f\right|V\nonumber\\ \Leftrightarrow V &= -\frac{\left|f\right|R}{2} \pm \sqrt{\frac{f^2R^2}{4} + R\frac{1}{\rho}\left|\nabla_hp\right|}. \end{align} \]

Because $V = \left|\mathbf{v}_h\right| > 0$ only the positive sign comes into question. In this case the wind speed is lower than the geostrophic wind speed, one speaks of subgeostrophic wind. The same applies in the anticyclonic case

\[ \begin{align} \frac{V^2}{R} &= -\frac{1}{\rho}\left|\nabla_hp\right| + \left|f\right|V\nonumber\\ \Leftrightarrow V^2 - \left|f\right|VR + R\frac{1}{\rho}\left|\nabla_hp\right| &= 0 \nonumber\\ \Leftrightarrow\frac{V^2}{R} &= -\frac{1}{\rho}\left|\nabla_hp\right| + \left|f\right|V\nonumber\\ \Leftrightarrow V &= \frac{\left|f\right|R}{2} \pm \sqrt{\frac{f^2R^2}{4} - R\frac{1}{\rho}\left|\nabla_hp\right|}. \end{align} \]

In this case the wind is supergeostrophic. Only the negative sign comes into play here, since in the limit case $\left|\nabla_hp\right| = 0$ no wind $V = R\left|f\right|$ should blow. The expression under the root is positive, so

\[ \begin{align} \left|\nabla_hp\right| \leq \frac{\rho f^2R}{4}. \end{align} \]

Anticyclones have a weaker gradient near the core; there is no such limitation for cyclones. This is because

\[ \begin{align} \frac{\rho f^2R}{4} \sim 10\text{ }\frac{\text{hPa}}{\text{1000 km}} \end{align} \]

actually significant on the synoptic scale.

13.10.3 Frictional wind

If you take a pressure gradient acceleration

\[ \begin{align} P \coloneqq \frac{1}{\rho}\left|\nabla p\right| \end{align} \]

which points in the y-direction and additionally a friction force $-\mu\mathbf{v}_h$ and sets the acceleration equal to zero, you get

\[ \begin{align} 0 &= fV\sin\left(\psi\right) - \mu V\cos\left(\psi\right) \stackrel{\psi\in\left[0, \pi/2\right]}{\Rightarrow} \psi=\arctan\left(\frac{\mu}{f}\right),\\ 0 &= P - fV\cos\left(\psi\right) - \mu V\sin\left(\psi\right) \Rightarrow V = \frac{P}{f\cos\left(\psi\right) + \mu\sin\left(\psi\right)}, \end{align} \]

where $\psi$ is the angle at which the wind intersects the isobar. This is called frictional wind. The frictional wind therefore leads to overisobaric transport and thus counteracts the formation of extremes in the ground pressure field or promotes their dissipation. The annihilation effect on a low is estimated with $R \sim 500$ km and $\psi \sim$ 30$^\circ$ from by

\[ \begin{align} \frac{\partial p}{\partial t} \sim \frac{g}{\pi r^2}\frac{dm}{dt} \sim \frac{g}{\pi r^2}2\pi r H\rho V\frac{1}{2} = \frac{gV H \rho}{r} \sim 4\text{ hPa/hr} \end{align} \]

where the boundary layer height was estimated to be $H = 500$ m. The frictional wind dissipates a cyclone within hours, so it is a relevant effect on the synoptic scale. Divergences above the friction layer were not taken into account, and $\psi$ was assumed to be independent of height within the boundary layer. In reality, of course, this is not the case, but $\psi$ decreases with height, which leads to the formation of a spiral-like wind field, the so-called Ekman spiral.

13.10.4 Euler wind

The Euler wind is the increase in wind under the effect of a pressure gradient and the absence of Coriolis acceleration, i.e

\[ \begin{align} \mathbf{v}_h = -t\frac{1}{\rho}\nabla p. \end{align} \]

13.10.5 Cyclostrophic wind

The cyclostrophic wind is given when a pressure gradient applies a centripetal force:

\[ \begin{align} \left|\frac{\partial p}{\partial r}\right| = \frac{\rho V^2}{\left|R\right|} \end{align} \]

It applies in Tornados and Dust Devils.

13.11 Quasigeostrophy

Here we start from the hydrostatic adiabatic equation system, which was summarized in Sect. 13.7:

\[ \begin{align} \md{_h\mathbf{v}_h} + \omega\frac{\partial\mathbf{v}_h}{\partial p} &= -f\mathbf{k}\times\mathbf{v}_h - \nabla\Phi,\tag{13.205}\label{eq:hydrostat_1}\\ \frac{\partial\Phi}{\partial p} &= -\alpha,\tag{13.206}\label{eq:hydrostat_2}\\ \frac{\partial^2\Phi}{\partial p\partial t} + \left(u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y}\right)\frac{\partial\Phi}{\partial p} + \sigma\omega &= 0,\tag{13.207}\label{eq:hydrostat_3}\\ \nabla\cdot\mathbf{v}_h + \frac{\partial\omega}{\partial p} &= 0,\tag{13.208}\label{eq:hydrostat_4}\\ p\alpha &= R_dT.\tag{13.209}\label{eq:hydrostat_5} \end{align} \]

Now some simplifications need to be made first. The goal is the quasigeostrophic system of equations. This concept is applied exclusively to channels in the extratropics that are narrow enough for the $\beta $plane. One also approximates the Coriolis parameter $f$

\[ \begin{align} f = f_0 + \beta\left(y - y_0\right)\stackrel{y_0 \equiv 0}{=}f_0 + \beta y, \end{align} \]

see section 13.9. Then Eq. applies to the geostrophic wind. (13.187)

\[ \begin{align} \mathbf{v}_{h, g} = \mathbf{k}\times\frac{1}{f}\nabla\Phi = \mathbf{k}\times\frac{1}{f_0}\nabla\Phi - \mathbf{k}\times\frac{\beta y}{f_0^2}\nabla\Phi + \mathcal{O}\left(f''\right)\stackrel{\text{in 0. Ordnung}}{\to}\mathbf{v}_{h, g} = \mathbf{k}\times\frac{1}{f_0}\nabla\Phi.\tag{13.211}\label{eq:geostr_wind_vereinfacht} \end{align} \]

From this one can derive the thermal wind

\[ \begin{align} \frac{\partial\mathbf{v}_{h, g}}{\partial p} = -\frac{1}{f_0}\mathbf{k}\times\nabla\alpha. \end{align} \]

The relative vorticity $\zeta$ is now approximately replaced by the relative vorticity of the geostrophic wind $\zeta_g$, which according to Eq. (15.86) approximately

\[ \begin{align} \zeta_g = \frac{1}{f_0}\Delta\phi. \end{align} \]

is given. The absolute vorticity is approximated from this

\[ \begin{align} \eta_g \coloneqq f + \zeta_g = f_0 + \beta y + \zeta_g. \end{align} \]

With Eq. (13.221) one now simplifies the material derivation to

\[ \begin{align} \md{} &= \md{_h} + \omega\frac{\partial}{\partial p} = \frac{\partial}{\partial t} + \mathbf{v}_h\cdot\nabla_h + \omega\frac{\partial}{\partial p}\nonumber\\ &\to \md{^{(g)}} \coloneqq\md{_h^{(g)}} + \omega\frac{\partial}{\partial p} = \frac{\partial}{\partial t} + \mathbf{v}_{h, g}\cdot\nabla_h + \omega\frac{\partial}{\partial p}. \end{align} \]

This follows for the first law

\[ \begin{align} \frac{\partial}{\partial t}\left(\frac{\partial\Phi}{\partial p}\right) = -\mathbf{v}_{h, g}\cdot\nabla_h\left(\frac{\partial\Phi}{\partial p}\right) - \sigma\omega.\tag{13.216}\label{eq:first_theorem_qg} \end{align} \]

The vorticity equation Eq. (15.85) is repeated here:

\[ \begin{align} \frac{\partial\zeta}{\partial t} &= -v\beta - \left(f + \zeta\right)\nabla\cdot\mathbf{v}_h - \mathbf{v}_h\cdot\nabla\zeta - \omega\frac{\partial\zeta}{\partial p} + \mathbf{k}\cdot\left[\frac{\partial\mathbf{v}_h}{\partial p}\times\nabla\omega\right] \end{align} \]

According to Sect. 13.10.1, the geostrophic wind is an order of magnitude larger than the ageostrophic component, so one can calculate the horizontal advection with $\mathbf{v}_{h, g}$ and replace $\zeta$ by $\zeta_g$:

\[ \begin{align} \frac{\partial\zeta_g}{\partial t} &= -v_g\beta - \left(f + \zeta_g\right)\nabla\cdot\mathbf{v}_h - \mathbf{v}_{h, g}\cdot\nabla\zeta_g - \omega\frac{\partial\zeta_g}{\partial p} + \mathbf{k}\cdot\left[\frac{\partial\mathbf{v}_{h}}{\partial p}\times\nabla\omega\right] \end{align} \]

The values ​​in tab. 1.3 also result in SI units

• $\frac{\partial\zeta_g}{\partial t}\sim10^{-10}$,
• $\omega\frac{\partial\zeta_g}{\partial p}\sim10^{-11}$,
• $\mathbf{k}\cdot\left[\frac{\partial\mathbf{v}_h}{\partial p}\times\nabla\omega\right]\sim10^{-11}$.

Consequently, the rotation term and the vertical advection can also be neglected, as can the relative versus absolute vorticity in the rotation term, which leads to the so-called quasigeostrophic vorticity equation with $f\approx f_0$

\[ \begin{align} \frac{\partial\zeta_g}{\partial t} &= -\mathbf{v}_{h, g}\cdot\nabla\left(\zeta_g + f\right) - f_0\nabla\cdot\mathbf{v}_h = -\mathbf{v}_{h, g}\cdot\nabla\left(\zeta_g + f\right) + f_0\frac{\partial\omega}{\partial p}.\tag{13.219}\label{eq:vorticity_p_qg} \end{align} \]

At this point, a current function is introduced to simplify things

\[ \begin{align} \psi \coloneqq\frac{\Phi}{f_0} \end{align} \]

one, so follow

\[ \begin{align} \mathbf{v}_{h, g} &= \mathbf{k}\times\nabla\psi, \tag{13.221}\label{eq:geostr_wind_stream}\\ \zeta_g &= \Delta_h\psi. \end{align} \]

From this it follows for the vorticity equation Eq. (13.219) and the First Law Eq. (3.12)

\[ \begin{align} \Delta\frac{\partial\psi}{\partial t} &= -\mathbf{v}_{h, g}\cdot\nabla\left(\Delta\psi + f\right) + f_0\frac{\partial\omega}{\partial p}, \tag{13.223}\label{eq:vorticity_p_qg_mod}\\ \frac{\partial}{\partial t}\left(\frac{\partial\psi}{\partial p}\right) &= -\mathbf{v}_{h, g}\cdot\nabla\left(\frac{\partial\psi}{\partial p}\right) - \frac{\sigma}{f_0}\omega\nonumber\\ \Rightarrow\frac{f_0^2}{\sigma}\frac{\partial^2}{\partial p^2}\left(\frac{\partial\psi}{\partial t}\right) &= -\frac{f_0^2}{\sigma}\mathbf{v}_h\cdot\nabla\left(\frac{\partial^2\psi}{\partial p^2}\right) - f_0\frac{\partial\omega}{\partial p} \end{align} \]

The last implication neglected the p-dependence on $\sigma$ and will continue to do so. Since $\sigma$ is related to stratification, this is a bad assumption, as of course in reality stratification is highly location dependent within the troposphere. Nevertheless, this assumption can be made here because the relevant phenomena are not filtered by this assumption. Furthermore, it was

\[ \begin{align} \frac{\partial\mathbf{v}_{h, g}}{\partial p}\cdot\nabla\left(\frac{\partial\psi}{\partial p}\right) \stackrel{\href{#eq:geostr_wind_stream}{\text{Glg. (13.221)}}}{=}\left(\mathbf{k}\times\nabla\frac{\partial\psi}{\partial p}\right)\cdot\nabla\left(\frac{\partial\psi}{\partial p}\right) = 0 \end{align} \]

used. Now the two equations are added

\[ \begin{align} \frac{\partial}{\partial t}\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right) + \mathbf{v}_{h, g}\cdot\nabla\left(f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2}\right) = 0.\tag{13.226}\label{eq:tendency_eq_pre} \end{align} \]

The size

\[ \begin{align} q_g \coloneqq f + \Delta\psi + \frac{f_0^2}{\sigma}\frac{\partial^2\psi}{\partial p^2} \end{align} \]

is called quasigeostrophic potential vorticity or also pseudopotential vorticity, so Eq. (13.226) as

note down. This is the so-called tendency equation. $\omega$ can be diagnosed from this by first applying the horizontal Laplace operator $\Delta_h$ to Equation. (13.216) to:

\[ \begin{align} \Delta\frac{\partial}{\partial p}\frac{\partial\psi}{\partial t} = -\Delta_h\left[\mathbf{v}_{h, g}\cdot\nabla\left(\frac{\partial\psi}{\partial p}\right)\right] - \frac{\sigma}{f_0}\Delta\omega.\tag{13.229}\label{eq:omega_eq_deriv_1} \end{align} \]

Now Eq. (13.219) differentiated according to $p$:

\[ \begin{align} \frac{\partial}{\partial p}\Delta\frac{\partial\psi}{\partial t} = -\frac{\partial}{\partial p}\left[\mathbf{v}_{h, g}\cdot\nabla\left(\Delta\psi + f\right)\right] + f_0\frac{\partial^2\omega}{\partial p^2}\tag{13.230}\label{eq:omega_eq_deriv_2} \end{align} \]

Now subtract Eq. (13.229) from (13.230) and gets

\[ \begin{align} 0 &= \Delta\left[\mathbf{v}_{h, g}\cdot\nabla\left(\frac{\partial\psi}{\partial p}\right)\right] + \frac{\sigma}{f_0}\Delta\omega - \frac{\partial}{\partial p}\left[\mathbf{v}_{h, g}\cdot\nabla\left(\Delta\psi + f\right)\right] + f_0\frac{\partial^2\omega}{\partial p^2}\nonumber \end{align} \]

This is the $\omega-$equation. This formalism is not globally applicable due to the linear expansion of $f$ contained therein; the performance lies in the fact that it can be used to understand baroclinic Rossby waves and baroclinic instabilities within zonal channels of limited extent in the extratropics, which is the theoretical basis of cyclogenesis.

13.12 Semigeostrophy

The semigeostrophic assumption is a relaxation of the quasigeostrophic assumption. In contrast to this, it leads to a globally applicable system of equations. As part of the solution spectrum, surfaces in the space of particular baroclinicity, so-called fronts, are obtained.

The starting point is the hydrostatic equation system derived in Section 13.7, which is noted again here:

\[ \begin{align} \frac{\partial u}{\partial t} - \mathbf{v}_h\cdot\nabla u - \omega\frac{\partial u}{\partial p} - fv + \frac{\partial\Phi}{\partial x} &= 0\\ \frac{\partial v}{\partial t} - \mathbf{v}_h\cdot\nabla v - \omega\frac{\partial v}{\partial p} + fu + \frac{\partial\Phi}{\partial y} &= 0\\ \frac{\partial^2\Phi}{\partial t\partial p} + \mathbf{v}_h\cdot\nabla\frac{\partial\Phi}{\partial p} + \frac{1}{\rho^2g^2}N^2\omega &= 0\\ \frac{\partial\omega}{\partial p} + \nabla\cdot\mathbf{v}_h&= 0,\\ \frac{\partial\Phi}{\partial p} + \frac{1}{\rho} = 0. \end{align} \]

Diabatic terms were omitted.

In the semigeostrophic approximation only the horizontal momentum equation is modified. is a basic requirement

\[ \begin{align} N_\text{Ro} = \frac{U}{Lf} \ll 1, \end{align} \]

so that the material acceleration is small compared to the Coriolis acceleration [10]. $N_\text{Ro}$ is the value in Eq. (13.3) defined Rossby number.

The semi-geostrophic system of equations is thus summarized as:

This is a system of equations for the prognostic variables $u$, $v$, $\omega$, $\Phi$, $\theta$. Note that $u_g$ and $v_g$ are diagnostic variables, although they have fixed time derivatives.