Abstract
Chapter 4 was devoted to the derivation and presentation of the governing equations of fluid mechanics and thermodynamics as they apply to fluid bodies under motion. The intention was to build a basic understanding of the mathematical description of the physical laws of balances of mass, momenta and energy in a form sufficiently general to all situations which one could possibly encounter in applications of physical limnology needed for this book.
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Notes
- 1.
More specifically, Coriolis and centripetal accelerations together prevent such a rest state. When centripetal accelerations are omitted, rest states are admissible.
- 2.
- 3.
From a point of view of direct application, it is obvious that determination of the directions of steepest descent, orthogonal to the bathymetric lines will provide a first information about the likely routes which such heavy littoral water might take when it dives to larger depths.
- 4.
Cabbeling, generally, is a physical process that is caused by the nonlinear terms in the expression of the density as a function of salinity and temperature measured at constant pressure. In physical oceanography, more specifically, cabbeling is a phenomenon that occurs when two water masses with identical densities but different temperatures and salinities mix to form a third water mass with a greater density than either of its constituents. This densification upon mixing is thought to cause the mixed fluid to flow downwards, away from the zone of mixing, and so will allow new source fluids to come in contact. For further reading, see, e.g. [29].
- 5.
Thermobaric effect – influence of the pressure on the coefficient of thermal expansion of water.
- 6.
We select here a rectangular basin with constant depth to have the same horizontal area of the lake at all depths. This allows in the ensuing analysis to work with water columns only and to omit the influence of the variable lake area with depth. The more complicated case may be treated as an exercise.
- 7.
Whenever estimates are made, we shall set g equal to 10 m s−2 rather than 9.81 m s−2. This will henceforth no longer be mentioned.
- 8.
Whether a storm is strong or exceptional depends on the regional meteorological conditions and the number of recurrences. An objective scale is the Beaufort’s scale , according to which wind speeds define the following conditions:
$$\begin{array}{ll}5-8 \,\mathrm{m\,s}^{-1},&\quad \mathrm{moderate wind},\\ 10-20\,\mathrm{m\,s}^{-1 },&\quad \mathrm{strong wind}, \\ 20-30\, \mathrm{m\,s}^{-1},&\quad \mathrm{storm},\\ >30\,\mathrm{m\,s}^{-1},&\quad \mathrm{hurricane}. \end{array}$$With this scale, a lake in a mountainous region would seldom be subject to a storm.
- 9.
The times t hom can be calculated according to \(t_\mathrm{hom} = \Delta \Pi_\mathrm{hom}/P\), however here the results are the same if \(\Delta \pi_\mathrm{hom}/P\) is evaluated instead. This is so because we assumed uniform wind and uniform surface water velocity.
- 10.
Formula (8.22) treats \(\Delta H_{\mathrm{E}}\) and \(\Delta H_{\mathrm{H}}\) analogously as if both were positive even though for (summer) stratification mixing leads to a temperature drop in the epilimnion for which a thermal contraction is expected. This is the case provided the epilimnion temperature is above \(T_*=4^{\circ}\mathrm{C}\). The sign of \(\Delta H_{\mathrm{E}}\) will decide whether under given conditions a thermal expansion (positive) or contraction (negative) will arise.
- 11.
This topic is a fashionable subject in elementary physics courses of College Physics, see, e.g., [18] or in courses on Environmental Fluid Mechanics, Meteorology, etc. The influence of friction, expressed in a hydrodynamic drag is, however, generally not treated.
- 12.
Accounting also for the compressibility, the buoyancy frequency is
$$N^2 = \frac{g}{\rho} \left( - \frac{\mathrm{d} \rho}{\mathrm{d} z} + \frac{g^2}{c^2} \right),$$where c is the speed of sound. This expression is useful for estimation of stability of the stratification in homogeneous layers in a lake interior.
- 13.
Note however, through the ratio \(A/V\) this constant depends on the geometry of the particle.
- 14.
\(\mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\,\,\sin\theta\).
- 15.
- 16.
Thermistor chains are usually 10, 20 or 50 m long and have 11 thermistors which are 1, 2 or 5 m apart.
- 17.
The dynamically important quantity is in fact the density distribution and therefore the distribution of the isopycnal surfaces rather than the isotherm–depth–time series. In a situation in which the thermal equation of state has the form \(\rho = \hat{\rho}(T,s)\), where s is salinity, measurements of temperature and salinity–time series will yield the density–time series from which the isopycnal time series can be constructed.
- 18.
This is an exact statement for an incompressible material for which ρ cannot depend on pressure. However, water is only nearly incompressible. So, (8.61) is an approximation for the case that the thermal equation of state does not depend on pressure.
- 19.
A proof of the theorem can be found in many books on linear algebra – inner product spaces – or in books on mathematical physics, e.g. [6].
- 20.
We omit the index n, because only a single mode does exist in this case, as will be demonstrated shortly. Furthermore, here we prefer to use the dimensional forms of the equations.
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Hutter, K., Wang, Y., Chubarenko, I.P. (2011). The Role of the Distribution of Mass Within Water Bodies on Earth. In: Physics of Lakes. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15178-1_8
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