13 Important approximations

In meteorology and oceanography there are many difficult approximations with detailed and complex assumptions. This chapter explains the most common 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 one wants to investigate a phenomenon, one always first chooses the simplest possible system of equations.

13.1.2 Dimensionless numbers

To compare different terms in the governing equations, one introduces dimensionless numbers.

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 one has

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

Synoptic flows 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 Sect. 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, one can simply use spherical coordinates for this; the orography can then 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. For the angle $\varphi$ that the ellipsoid encloses with a spherical surface, one has

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

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 one wants to take the eccentricity of the gravity field into account, one 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 system of equations, Eqs. (8.104) - (8.105), under the SGA becomes

$\mathbf{g}$ must be irrotational as a gradient field,

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

According to Eq. (B.114), one has

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

Thus, in the SGA, one has

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

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

The SGA is a relatively weak approximation, which nevertheless greatly simplifies the system of equations. It is therefore almost always appropriate for analytical considerations. For models, it can be dropped, see Sect. 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. Substituting this into Eq. (B.98), one obtains

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

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

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

In the shallow atmosphere, for the velocity resulting from the earth's rotation, one has

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

With Eq. (13.17), one obtains

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

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.

Substituting Eq. (13.15) into Eq. (B.90), one obtains

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

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

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

Since the density of the atmosphere is neglected when calculating $\mathbf{g}$, one has, 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 made,

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 allows Eqs. (13.25) - (13.26) to be written more concisely as

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

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 introduced in [1]. To motivate it, one first considers the shallow-atmosphere equations in the form given in Sect. 10.1.2:

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

Only Eq. (13.37) is modified. For this purpose, one introduces a background state $\left(\newoverline{\Pi}\left(z\right), \newoverline{\theta}\left(z\right)\right)$; deviations from it are denoted, as usual, by primed quantities. Under the assumption $\Pi' \ll \newoverline{\Pi}$, Eq. (13.37) can be written as

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

The pseudo-incompressible approximation now consists in neglecting the term $\frac{c^{(V)}}{R_d\Pi}\md{\Pi}$ in Eq. (13.39), i.e. in assuming

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

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

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

From this, the chain rule gives

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

Substituting this into Eq. (13.40) yields

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

This leads to the compact formulation

of 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

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

Substituting Eqs. (13.45) - (13.47) here, one obtains

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

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

13.5.1 Horizontal momentum equation

If one replaces the density in the prefactor of the acceleration (including the Coriolis acceleration) by its mean value, one obtains

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

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 one projects Eq. (13.55) onto $\mathbf{k}$, one obtains

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

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

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

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

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

From this it follows

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

Substituting this into Eq. (13.59), one obtains

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

From Eq. (13.61), it follows

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

Expanding this to first order around $\left(\rho_0, p_0\right)$, one obtains

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

Thus, neglecting the approximate-equality sign, one has

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

Define the buoyancy $b$ by

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

the vertical momentum equation of the anelastic approximation reads

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 one substitutes Eq. (13.45) into the continuity equation, one obtains

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

Neglecting the fluctuation $\rho'$ here, one obtains

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. For it, write

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

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 $D_\psi$ corresponding to $\psi$ 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, one assumes a homogeneous system. In this case, one can write the thermal equation of state in the form

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

This can be expanded into a Taylor series about the expansion point $\left(p, T\right)^T = \left(p_m, T_m\right)^T$:If one defines $\rho_m, p_m, T_m$ as the means of the respective quantities, then, owing to the nonlinearity of the equation of state, in general $\rho_m \not= \rho\left(p_m, T_m\right)$. It therefore makes sense to define $\rho_m \coloneqq \rho\left(p_m, T_m\right)$.

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

Here, the definitions

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

were used. For ideal gases, with $\rho = \frac{p}{R_sT}$, one has

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

For fluids, one can estimate

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

Substituting this into Eq. (13.83), also for the second derivatives, one obtains, neglecting the terms of third and higher order,

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

Owing to Eqs. (13.80) and (13.81), one can approximate the right-hand side by

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

Thus, neglecting the approximate-equality sign (to first order in the perturbations $\psi - \psi_m$ of the thermodynamic variable $\psi$), one has

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

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

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

Substituting this into the momentum equation, one obtains

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

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), one can approximate

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

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

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

Substituting Eq. (13.80) here, one obtains

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

One can use the momentum equation Eq. (13.98) to simplify it a little further. 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), one can estimate

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

$H'$ 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 estimate

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

so one also has

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

This means that in Eq. (13.103), one can approximate

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

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 one multiplies Eq. (9.10) by the density $\rho$, one obtains

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

where $q^{(V)}$ is the heat power density. Owing to Eq. (13.77), one has

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

Substituting this into Eq. (13.111), one obtains

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

where in the prefactor of the temperature tendency the density $\rho$ was replaced by the mean density $\rho_m$. To first order in the deviations from the averaged quantities, the compression term can be simplified to

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

With Eq. (13.91), it follows, neglecting the approximate-equality 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 here, one approximates 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), to first order in the deviation from the mean density, one has

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

Thus, neglecting the approximate-equality sign, one has

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

Substituting this into Eq. (13.113), one obtains

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

This is the temperature equation of the Boussinesq approximation:

For the ideal gas, with Eqs. (13.86) - (13.87), one has

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

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 to 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 velocity. This must be zero, since otherwise it becomes the rule that a particle disappears into the ground or into space. Thus the implication holds

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

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

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

with $\alpha \coloneqq \frac{1}{\rho}$ the specific volume. From the equation of state, $\alpha = \frac{R_dT}{p}$ follows in a dry atmosphere. That is 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, in general, reads

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

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, $\frac{\partial}{\partial z} = \frac{\partial p}{\partial z}\frac{\partial }{\partial p}$. It thus follows

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

Using Eq. (12.7), one obtains

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

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

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. Scaling 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 one wants to transform the equations of motion for shallow geofluids, Eqs. (13.31) - (13.32), only the total derivatives in the p-system need to be written on the left-hand side. The pressure gradient takes a little more work. With Eq. (12.7) and the hydrostatic approximation, it follows

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

Thus one has

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

and analogously in the $y$-direction. Thus the horizontal momentum equations in the p-system become

13.7.3 Temperature equation

If one differentiates the equation of state $p\alpha = R_dT$ with respect to 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), neglecting the mass source density, it follows

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

Thus one obtains

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

With the equation of state follows

this defines the stability parameter $S_p$. One obtains

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

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

Substituting this 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

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

By Eq. (13.122), this is connected, via

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

with the Brunt-Väisälä frequency $N$. One obtains

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 can be seen, $\omega$ is a diagnostic quantity. $\alpha$ can be determined from $\Phi$ by means of Eq. (13.124); subsequently, the temperature can also be calculated diagnostically via $T = \frac{p\alpha}{R_d}$.

13.7.5 Hydrostatic background state

Even if one does not make the hydrostatic assumption, one can introduce a basic state $\{\newoverline{\alpha}, \newoverline{p}\}$, with $\alpha \coloneqq \frac{1}{\rho}$ the specific volume, that satisfies Eq. (13.122), and write the actual quantities $\{\alpha, p\}$ as a superposition of the basic state with deviations $\{\alpha', p'\}$, i.e.,

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

Then one has

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

The Earth's gravity field has a complicated shape, which one approximates in various stages. First of all, it is assumed that the gravity field is radially symmetric with a homogeneous magnitude of the gravity vector. If one uses 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 hydrostatic basic equation reads, by the chain rule,

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

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 one wants to take into account the height dependence of the gravitational acceleration, one first uses, as the formula for the gravitational potential,

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

with $\Phi_0 \coloneqq GM/a$, where $G$ is Newton's gravitational constant and $M$ the 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, for the generalized vertical coordinate geopotential height $z_g$, that

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

holds. From this it further follows

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

13.8 Barotropy

One speaks of barotropy when the contour surfaces of the pressure field and the density field coincide. In a dry atmosphere this is equivalent to the fact that the temperature surfaces and the pressure surfaces are equal. In reality, the baroclinicity 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.

Differentiating the $x$-component of Eq. (13.132) with respect to $p$ under the assumption of barotropy, one obtains

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

One also has

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

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 one sees that if the vertical shear vanishes globally at one point in time, it does so for all points in time. If the vertical shear vanishes, then the horizontal divergence, and thus, by Eq. (13.129), $\frac{\partial\omega}{\partial p}$, are also height-constant. 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 the shallow water equations (SWEs). To derive them, one starts from Eqs. (13.31) - (13.32), assumes a homogeneous density, and moreover ignores friction. The continuity equation thereby becomes

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

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 $b = b\left(x, y\right)$. By integrating the hydrostatic basic equation, one thus obtains

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

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 motion $\left(\frac{\partial u}{\partial z}, \frac{\partial v}{\partial z}\right)^T$ 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, which made the integration in Eq. (13.165) possible, gives the shallow water equations their name. At great depths, this can clearly no longer be assumed. In Sect. 16.5.2, this assumption is relaxed. When dealing with waves, one can use the shallow water equations if

\[ \begin{align} \text{wavelength} &\gg \text{depth},\\ \text{wave height} &\ll \text{depth} \end{align} \]

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:

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

At the surface, one has

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

Substituting this into Eq. (13.165), one obtains

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

This results in the following system of equations:

13.8.1.1 Linearization

Assuming a homogeneous subsurface $b$ and a mean depth $D$ on which a perturbation $d$ 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 Expansions of the Coriolis parameter in plane geometry

For the Coriolis parameter $f$ as a function of latitude $\varphi$, one has

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

Taylor-expanding this at a latitude $\phi_0$, one obtains

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

where the terms of fourth and higher order were no longer written. With

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

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

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

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 $2B$, centered at the latitude $\varphi_0$. The curvature terms are neglected, which is why one speaks of planes.

13.9.1 $f$-plane

Here one makes the approximation

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

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

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

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

13.9.2 $\beta-$plane

Here one makes the approximation

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

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

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

hold.

13.10 Equilibrium winds

13.10.1 Geostrophy

Scaling the horizontal momentum equation on the synoptic scale.

Term	Order of 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 $N_\text{Ro}$, one has

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

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).

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

Solving this for the wind velocity, one obtains

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

So the geostrophic wind is

The geostrophic wind speed thus decreases with increasing magnitude of the latitude; at the equator it tends to infinity. It must therefore be questioned up to which latitudes geostrophy is a good approximation. This question is examined in more detail in Sect. 13.9.2.

Forming the scalar 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), one obtains

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

Using the scales from Tab. 1.3, it 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 baroclinicity. From Eq. (13.187), it 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 surface. Eq. (13.191) is the thermal wind equation.

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

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

with a weighted layer-mean 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

For the gradient wind, one assumes that the Coriolis force and the pressure gradient together provide the centripetal force 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.,

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

Since $V = \left|\mathbf{v}_h\right| > 0$, only the positive sign is possible. In this case, the wind speed is lower than the geostrophic wind speed; one speaks of subgeostrophic wind. In the anticyclonic case, one has analogously

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

In this case the wind is supergeostrophic. Here, only the negative sign is possible, since in the limiting 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 limitation is actually significant on the synoptic scale, since

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

13.10.3 Frictional wind

If one assumes a pressure gradient acceleration

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

pointing in the $y$-direction, and additionally a friction force $-\mu\mathbf{v}_h$, and sets the acceleration equal to zero, one obtains

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

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 destructive effect on a low is estimated, with $R \sim 500$ km and $\psi \sim$ 30$^\circ$, 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 the 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 holds in tornadoes and dust devils.

13.11 Quasigeostrophy

Here one starts from the hydrostatic adiabatic system of equations, which was summarized in Sect. 13.7.4:

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

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 $f$ to

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

cf. Sect. 13.9. Then, for the geostrophic wind, Eq. (13.187) gives

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

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, by Eq. (15.86), is approximately given by

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

For the absolute vorticity, one approximates from this

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

With Eq. (13.211), one now simplifies the material derivative to

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

From this, for the first law, it follows

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

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 additionally yield, 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, for simplification, one introduces a stream function

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

from which 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, for the vorticity equation Eq. (13.219) and the first law Eq. (3.12), it follows

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

The last implication neglected the $p$-dependence of $\sigma$ 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,

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

was 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 quantity

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

is called quasigeostrophic potential vorticity or also pseudopotential vorticity, with which Eq. (13.226) can be written as

This is the so-called tendency equation. $\omega$ can be diagnosed from this; to this end, one first applies the horizontal Laplace operator $\Delta_h$ to Eq. (13.216):

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

Now Eq. (13.219) is differentiated with respect 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 one subtracts Eq. (13.229) from (13.230) and obtains

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

This is the $\omega-$equation. This formalism is not globally applicable due to the linear expansion of $f$ contained therein; its power 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 system of equations derived in Sect. 13.7, which is written out 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. The basic requirement is

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

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

The semigeostrophic system of equations thus reads, in summary:

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.