Abstract
In the chapter Dynamics, and Synoptic, the basic laws for describing the kinematic and dynamic behavior of tropospheric flows are presented and discussed. The conservation equations for momentum (Newton’s 2nd law), total mass (equation of continuity), dry air, water substances, trace constituents, and energy (1st principle of thermodynamics) are presented and explained, where inertial frames and moving frames rotating with the Earth are considered. This presentation includes different kinds of coordinate systems. Simplifications like the hydrostatic and geostrophic approximations are related to scaling considerations (scale analysis). Balanced curved flows, streamlines and trajectories are explained as well. Circulation and vorticity principles are discussed to analyze rotational flows. This part includes, for instance, the balance equation for vorticity and the distinction between absolute and relative vorticity. Wave analysis is explained by examples like gravity waves and Rossby waves. Principles of Ekman’s physics of the atmospheric boundary layer (ABL) are presented to point out the effects of turbulent motion. The chapter also encompasses how principles of dynamics and kinetic as well as numerical weather prediction model results are used in weather forecasting.
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Notes
- 1.
This chapter pre-assumes basic knowledge of calculus. A short review of the fundamental concepts relevant for understanding of this chapter is given in Appendix A.
- 2.
Due to the rotation of the Earth a classroom with its center at the pole turns around once a day. At the equator, a classroom makes a free trip at its distance from the Earth’s center around the rotation axis.
- 3.
Carl-Gustaf Arvid Rossby, Swedish-U.S. meteorologist, 1898–1957. He was the first to explain the large-scale motions of the atmosphere in terms of fluid mechanics.
- 4.
Hans Ertel, German natural scientist and pioneer in geophysics, meteorology and hydrodynamics, 1904–1971.
- 5.
The inertial system is a coordinate system realized by coordinates that are fixed in space.
- 6.
\(\varOmega = \frac{2\pi } {\mathit{siderianday}} = \frac{2\pi } {86,164\,\mathrm{s}} = 7.292 \cdot 10^{-5}\) s−1. Note that a sidereal (or star) day has 23 h, 56 min and 4.09 s. The rotational velocity of the Earth with respect to a fixed point in the space is slightly shorter than a day (24 h).
- 7.
The triple vector product of three vectors, a, b, and c is given by \(\mathbf{a} \times \left (\mathbf{b} \times \mathbf{c}\right ) = \left (\mathbf{a} \cdot \mathbf{c}\right )\mathbf{b} -\left (\mathbf{a} \cdot \mathbf{b}\right )\mathbf{c}\). Furthermore, \(\boldsymbol{\varOmega }\) is spatially constant, \(\nabla \cdot \mathbf{r} = 3\) and \(\nabla \mathbf{r} = \mathbf{E}\), where E is the identity tensor.
- 8.
In meteorology, the x-direction of the coordinate system is defined positive from west to east, the y-direction points from the south to the north and the z-direction is directed upwards positive. Winds from west/south have a positive sign while winds coming from east/north have a negative sign.
- 9.
This phenomenon often occurs in jet streams.
- 10.
The gravitational constant was first determined by Cavendish in 1798.
- 11.
There are four fictitious forces, rectilinear acceleration, centrifugal force, Coriolis force, and the Euler force.
- 12.
Jean le Rond d’Alembert, French mathematician, mechanician, physicist and philosopher, 1717–1783.
- 13.
To visualize the curved path on a rotating surface, try to draw a straight line on a piece of paper while a classmate rotates it slightly.
- 14.
In meteorology, the basic equations are written commonly without explicitly writing the mass. The density of air is close to 1 kg m−3 in the lower atmosphere. Using a unit volume of 1 m3 leads to the unit mass of air being approximately 1 kg.
- 15.
General circulation models (GCMs) and NWP models usually base on the Euler equation and re-introduce the friction at the atmosphere-surface interface and in the atmospheric boundary layer indirectly by parameterizations.
- 16.
Examples of analytical solutions can be found in textbooks on atmospheric dynamics or numerical modeling. These solutions are used to test numerical models during their development phase.
- 17.
Advection is the horizontal movement of air.
- 18.
Christophorus Henricus Diedericus Buys Ballot, Dutch chemist and meteorologist, 1817–1890.
- 19.
Rearranging for v and inserting in \(\frac{dv} {dt} = -fu -\frac{uv} {r}\) provides \(\begin{array}{l} \quad -\frac{1} {\rho } \frac{\partial p} {\partial r} + \frac{fm} {r} -\frac{f^{2}r} {2} + (\frac{m^{2}} {r^{2}} -\frac{2mf} {2} + \frac{f^{2}r^{2}} {4} )/r \\ = -\frac{1} {\rho } \frac{\partial p} {\partial r} + \frac{fm} {r} - 2\frac{f^{2}r} {4} + \frac{m^{2}} {r^{3}} -\frac{mf} {r} + \frac{f^{2}r} {4} \\ = -\frac{1} {\rho } \frac{\partial p} {\partial r} -\frac{f^{2}r} {4} + \frac{m^{2}} {r^{3}}\\ \end{array}\)
- 20.
\( \begin{array}{lll} \frac{dm} {dt} = r\frac{dv} {dt} + v\frac{dr} {dt} + \frac{f2r} {2} \frac{dr} {dt} \\ = r\frac{dv} {dt} + (v + fr)\frac{dr} {dt} \\ = r(-fu -\frac{uv} {r} ) + (v + fr)\frac{dr} {dt} \\ = -fru - uv + uv + fru = 0 \end{array} \)
- 21.
Here numbers are used to address the different phase of water and dry air as often done in theoretical meteorology to reduce the writing burden.
- 22.
Herein it is assumed that moist air can be considered as an ideal gas. In the nomenclature introduced above, the computed gas constant for the gas mixture dry air, R d , is denoted to as R 0, ρ is the density of air, and T v = T(1 + 0. 61q 1) is the virtual temperature, and Θ is the potential temperature using the nomenclature of theoretical meteorology and ABL physics (Chap. 2).
- 23.
The unit vectors in Cartesian (i, j, k) and spherical coordinates (i λ , j ϕ , k r ) differ. Only tensors of same rank, and tensors and vectors belonging to the same coordinate systems can be added.
- 24.
Osborne Reynolds, British physicist, 1842–1912.
- 25.
Joseph Valentin Boussinesq, French mathematician and physicist, 1842–1929.
- 26.
Osborne Reynolds, British mathematician and engineer, 1842–1912.
- 27.
For low Rossby numbers we can extend the Boussinesq approximation to large vertical scales of motion when we use modified pressure and potential density.
- 28.
Sound waves may not be ignored in meteorological models that aim at investigation of noise annoyance.
- 29.
Ernst Heinrich Wilhelm Schmidt, German engineer, 1892–1975.
- 30.
Jean Baptiste Joseph Fourier, French mathematician and physicist, 1768–1830.
- 31.
Adolf Eugen Fick, German physician, 1829–1901.
- 32.
Fritz Herbert, German meteorologist, born 1944.
- 33.
Lars Onsager, Norwegian-American physical chemist, 1903–1976.
- 34.
Pierre Curie, French physicist, 1859–1906.
- 35.
Ilya Romanovich Prigogine, Belgian physical chemist, 1917–2003.
- 36.
Julius C. Rotta, German engineer, 1912–2005.
- 37.
Warren Kendall Lewis, American chemical engineer, 1882–1975.
- 38.
Andrey Nikolaevich Kolmogorov, Russian mathematician, 1903–1984.
- 39.
Alexander Mikhailovich Obukhov, Russian physicist, 1918–1989.
- 40.
Werner Karl Heisenberg, German physicist, 1901–1976.
- 41.
Carl Friedrich von Weizsäcker, German physicist, 1912–2007.
- 42.
Vagn Walfrid Ekman, Swedish oceanographer, 1874–1954.
- 43.
Ludwig Prandtl, German aeronautical engineer, 1875–1953.
- 44.
Originally Ekman derived this relationship for the ocean by using the kinematic viscosity of water instead of the eddy diffusivity for momentum. Ekman also adduced some arguments in favor of using his results to describe the variation of wind speed with height over land. Later, independently of each other, Akerblom in 1908 and Exner in 1912 realized Ekman’s idea. Exner was the first who combined both the planetary boundary layers of the ocean, and the atmosphere in a single dynamic system.
- 45.
Heinz Helmut Lettau, German-American meteorologist, 1909–2005.
- 46.
Alfred K. Blackadar, American meteorologist, born 1920.
- 47.
Jule Gregory Charney, Swedish-American meteorologist, 1917–1981.
- 48.
Arnt Eliassen, Norwegian meteorologist, 1915–2000.
- 49.
Andrei Sergeevich Monin, Russian physicist, 1921–2007.
- 50.
Charles Henry Brian Priestley, British meteorologist, 1915–1998.
- 51.
Grigory Isaakovich Barenblatt, Russian mathematician, born 1927.
- 52.
- 53.
Edgar Buckingham, American physicist, 1867–1940.
- 54.
Gabriel Cramer, Swiss mathematician, 1704–1752.
- 55.
Sir Geoffrey Ingram Taylor, British physicist, 1886–1975.
- 56.
This means that replacing Q 1 = z − d in the similarity hypothesis by \(Q_{1} =\varLambda _{P} =\kappa \; \left (z - d\right )\) would provide the same result.
- 57.
Sergej S. Zilitinkevich, Russian-Swedish hydrometeorologist, born 1936.
- 58.
The Businger-Dyer-Pandolfo relationship was later experimentally determined by Dyer and Hicks, Businger et al. and others, where their results mainly cover the stability range − 2 ≤ ζ < 0.
- 59.
O’KEYPS stands for the initials of various authors who proposed this formula (Obukhov, Kazansky and Monin, Ellison, Yamamoto, Panofsky, and Sellers).
- 60.
Hans A. Panofsky, German-American meteorologist, 1917–1988.
- 61.
John A. Dutton, American meteorologist.
- 62.
John L. Lumley, American mechanical and aerospace engineer, born 1930.
- 63.
James W. Deardorff, American meteorologist, born 1928.
- 64.
Dispersion and dissipation are physically different processes. In a dynamical system, dissipation denotes the concept where important mechanisms (e.g. waves, oscillations) lose energy with time because of friction, radiative cooling or turbulence. Energy is converted to heat for which the temperature of the system increases. When a wave loses amplitude, it dissipates.
- 65.
Be aware that in principle, zero pressure perturbation contradicts to non-zero buoyancy when mass is conserved.
- 66.
Hermann Ludwig Ferdinand von Helmholtz, German physician and physicist, 1821–1894.
- 67.
Vilhelm Friman Koren Bjerknes, Norwegian physicist and meteorologist, founder of the modern practice of weather forecasting, 1862–1951.
- 68.
Max Margules, Austrian mathematician, physicist, and chemist, 1856–1920.
- 69.
Sometimes UTC is abbreviated as UT.
- 70.
The same universal time is needed so that weather maps can be drawn.
- 71.
Material, concepts, ideas and problems of the following books and articles inspired this chapter. These sources are recommended for further reading.
References
Material, concepts, ideas and problems of the following books and articles inspired this chapter. These sources are recommended for further reading.
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Mölders, N., Kramm, G. (2014). Dynamics and Synoptic. In: Lectures in Meteorology. Springer Atmospheric Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-02144-7_6
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