Max H. Balsmeier
© Max H. Balsmeier 2020 - 2026
This work most likely contains errors and inaccuracies. It is under permanent revision. Some parts are incomplete or only headings have been inserted so far. Commercial use is prohibited. All rights reserved.
There are many good textbooks in the various branches of meteorology; in dynamics in particular, the body of literature is extensive and continually growing. Cloud physics and radiation likewise offer substantial standard works, alongside a number of books that are somewhat leaner in content but more explicitly designed for teaching. Yet, as is typical of textbooks – and of many lecture courses – some derivations are presented in simplified, or at least abridged, form. For the theoretically interested reader or listener, this can leave a lingering sense of doubt about the correctness of the theory. In addition, physical quantities are denoted in many different ways across the literature, so that literature searches – for example on derivations and the prerequisites of specific concepts – do not always produce clarity. This book is intended to close that gap. It should be understood neither as a pure textbook – where the verbal explanations would presumably have been somewhat less condensed – nor as a mere formula compendium, since such a work would not include derivations.
The reader should find answers to questions such as
How does one arrive at the various forms of the divergence equation in the pressure-coordinate system?,
Under what conditions is quasi-geostrophic theory an appropriate approximation?
and will encounter a mathematically rigorous presentation of the material (at least by the standards of a physicist), but not one reduced to didactic simplification; all assumptions and limitations of a concept are explicitly identified as such. The derivations are carried out in sufficiently small steps that a reader fully at ease with secondary-school mathematics should not need to perform supplementary calculations.
The scope includes dynamics, radiation, cloud microphysics, and numerical methods. Part I assembles the physical foundations needed to understand the atmosphere. Part II attempts to compile a system of equations that describes all atmospheric processes. The collected foundational knowledge is then applied to concrete subfields of atmospheric theory. Part III prepares theoretical tools for the treatment of air flows in a planetary atmosphere – the field referred to as dynamics. The following Part IV applies this theory to specific atmospheric and oceanic problems. By contrast, Part V addresses the additional domains of radiation and cloud microphysics. Numerical problems are treated in the sixth part. The seventh develops a dynamical core. Mathematical foundations are worked through in the appendix.
The SI system is used as the system of units.
| Symbol / name | meaning | value (where applicable) |
|---|---|---|
| $k_B$ | Boltzmann constant | $1.380649\cdot 10^{-23}$ JK$^{-1}$ [15] |
| N$_A$ | Avogadro constant | $6.0221409\cdot 10^{23}$ mol$^{-1} = 1$ [15] |
| $R = k_B\cdot N_A$ | universal gas constant | $8.314463$ Jmol$^{-1}$K$^{-1}$ |
| $R_s$ | specific gas constant | |
| $c^{(p)}$ | isobaric specific heat capacity | |
| $c^{(V)}$ | isochoric specific heat capacity | |
| subscript $d$ | refers to dry air | |
| subscript $v$ | refers to water vapour | |
| subscript $g$ | refers to the gaseous fraction of air | |
| $M_d$ | molar mass of dry air | $0.028964420$ kg/mol [22] |
| $M_v$ | molar mass of water | $0.01801527$ kg/mol [28] |
| $\omega$ | angular velocity of Earth's rotation | $7.292115\cdot 10^{-5}$ s$^{-1}$ [29] |
| $a$ | Earth's radius at the equator | $6{,}378{,}137.0$ m [40] |
| $1/\newtilde{f}$ | flattening | $298.257223563$ [40] |
| $\beta$ | obliquity of Earth's axis | $23.439279^\circ$ [29] |
| $S_0$ | solar constant | $1361$ W/m$^2$ [31] |
| $\Omega$ | angular velocity of Earth's revolution | $1.99099\cdot 10^{-7}\text{ s}^{-1}$ [31] |
| $M$ | mass of the Earth | $5.9723\cdot 10^{24}\:\text{kg}$ [31] |
| specific | per unit mass |
The following works are key references in their respective fields:
Collection of mathematical formulae: Abramowitz et al. [11]
Canon of Theoretical Physics: e.g. Fließbach [18], [19], [20], [21]
Fluid mechanics: Kundu [7]
Radiation transport: Chandrasekhar [14]
Cloud microphysics: Pruppacher and Klett [33]
Water surface waves: Massel [37]
| name | alternative name | length scale / m | time scale / s | typical phenomena |
|---|---|---|---|---|
| synoptic scale | large scale | $> 1\cdot 10^6$ | $> 1\cdot 10^5$ | Rossby waves, extratropical depressions |
| Meso$-\alpha-$scale | $2\cdot 10^{5}$ - $2\cdot 10^6$ | $2\cdot 10^{4}$ - $2\cdot 10^5$ | tropical cyclones, fronts | |
| Meso$-\beta-$scale | $2\cdot 10^{4}$ - $2\cdot 10^5$ | $2\cdot 10^{3}$ - $2\cdot 10^4$ | land-sea breeze circulation | |
| Meso$-\gamma-$ scale | storm scale, convective scale | $2\cdot 10^{3}$-$2\cdot 10^4$ | $2\cdot 10^{2}$-$2\cdot 10^3$ | convection, leeward waves, Kelvin-Helmholtz instability |
| microscale | $2\cdot 10^{-3}$-$2\cdot 10^3$ | $2\cdot 10^{-4}$ - $2\cdot 10^2$ | turbulence, flow around houses and trees etc. | |
| molecular scale | $< 2\cdot 10^{-3}$ | $< 2\cdot 10^{-4}$ | momentum and mass diffusion, radiation |
| quantity | order of magnitude |
|---|---|
| synoptic length scale $L$ | $10^6$ m |
| horizontal wind $u, v$ | $10^{1}$ ms$^{-1}$ |
| vertical wind $w$ | $10^{-2}$ ms$^{-1}$ |
| gravity $g$ | $10^{1}$ ms$^{-2}$ |
| characteristic height $H$ | $10^{4}$ m |
| time scale $T = L/u$ | $10^5$ s |
| Earth's radius $a$ | $10^7$ m |
| density $\rho$ | $10^0$ kgm$^{-3}$ |
| Coriolis parameter $f$ | $10^{-4}$ s$^{-1}$ |
| horizontal pressure fluctuation $\delta p$ | $10^{3}$ Pa |
| vertical pressure fluctuation $\delta p$ | $10^{5}$ Pa |
| Rossby parameter $\beta$ | $10^{-11}$ m$^{-1}$s$^{-1}$ |
| relative vorticity $\zeta$ | $10^{-5}$ s$^{-1}$ |
| horizontal divergence $\delta$ | $10^{-5}$ s$^{-1}$ |
| $p$-vertical velocity $\omega$ | $10^{-1}$ Pas$^{-1}$ |
The term scales refers to characteristic orders of magnitude. The goal of scale analysis is to simplify equations. Different scales are characterised by the different phenomena that typically occur on them. Tab. 1.3 provides an overview of the time and length scales commonly used in meteorology, their names, and typical associated phenomena. Some characteristic quantities of the synoptic scale are listed in Table 1.3.
| abbrev. | pronunciation | meaning |
|---|---|---|
| P | peta | $10^{15}$ |
| T | tera | $10^{12}$ |
| G | giga | $10^{9}$ |
| M | mega | $10^{6}$ |
| k | kilo | $10^{3}$ |
| h | hecto | $10^{2}$ |
| da | deca | $10^{1}$ |
| d | deci | $10^{-1}$ |
| c | centi | $10^{-2}$ |
| m | milli | $10^{-3}$ |
| $\mu$ | micro | $10^{-6}$ |
| n | nano | $10^{-9}$ |
| p | pico | $10^{-12}$ |
| f | femto | $10^{-15}$ |
| a | atto | $10^{-18}$ |