Wedderburn number

The Wedderburn number is a dimensionless number that is mainly used in physical limnology . It describes the size of upwelling in stratified lakes and is defined as:

${\displaystyle W={\frac {g'h_{e}^{2}}{u_{*}^{2}L}}}$,

where (with the spatial factor ) is the reduced gravity, the thickness of the epilimnion , the speed of shear stress and the length of the lake. The lake is described in a simplified manner by two layers: An upper layer with lighter, warm water (epilimnion) (density ), which rests on the heavier hypolimnion (density ). High Wedderburn numbers mean a stable water column, while low numbers mean a strong mix, especially on the shore zones. ${\displaystyle g'=g{\frac {(\rho _{e}-\rho _{h})}{\rho _{e}}}}$ ${\displaystyle g}$ ${\displaystyle h_{e}}$ ${\displaystyle u_{*}}$ ${\displaystyle L}$ ${\displaystyle \rho _{e}}$ ${\displaystyle \rho _{h}}$

motivation

In many stratified lakes, wind is one of the most important sources of physical energy. However, only a small part of the wind energy leads to direct vertical mixing of the water column. Instead, sustained winds often cause the thermocline to tilt : the water of the epilimnion is accelerated in the direction of the wind and transported to the leeward side, while on the windward side it is replaced by deep water in a buoyancy movement. The size of this shift is determined by the ratio of the stabilizing potential energy of the water column ( ) to the kinetic energy ( ), which is given in the bulk Richardson number ${\displaystyle E_{pot}}$ ${\displaystyle E_{kin}}$

${\displaystyle R_{B}={\frac {E_{pot}}{E_{kin}}}={\frac {g'h_{e}}{u_{*}^{2}}}}$

is expressed. Furthermore, the length of the fetch determines how effective the wind can act on the lake. Because of this, the ratio of the length of the lake to the thickness of the epilimnion becomes

${\displaystyle A={\frac {h_{e}}{L}}}$

with multiplied to the Wedderburn Number ${\displaystyle R_{B}}$

${\displaystyle W=AR_{B}={\frac {g'h_{e}^{2}}{u_{*}^{2}L}}}$

to obtain. The Wedderburn number also determines the angle of inclination of the thermocline: the smaller it is, the more it is deflected. At , the angle of inclination becomes so great that the deep water of the hypolimnion reaches the surface at one end of the lake. ${\displaystyle W\approx 1}$

meaning

The Wedderburn number is a way of quantifying mixing processes in lakes. At low temperature differences in the lake and high winds, the Wedderburn number is close to one, at low wind speeds and high temperature differences between the epi- and hypolimnion, on the other hand, it can well exceed the value of 100. Many biological processes in lakes depend on the mixture of the epilimnion. Heavier plankton species such as diatoms sink into deep water if they are not constantly stabilized by mixing. On the other hand, large amounts of zooplankton , which serve as food for fish larvae, can accumulate during quieter periods, for example . The mixture is also of great importance for the nutrient balance. With low Wedderburn numbers, nutrients can be transported from the deep water to the surface water, where they are available to organisms. A fundamental problem with the Wedderburn number is its simplistic representation of the lake as a two-layer system. One improvement is the lake number , which instead integrates the density across the entire water column.

Remarks

1. ↑ T. Shintani et al. A. (2010): Generalizations of the Wedderburn number: Parameterizing upwelling in stratified lakes . In: Limnology and Oceanography 55 (3), pp. 1377-1389.
2. ^ J. Imberger, P. Hamblin (1982): Dynamics of lakes, reservoirs and cooling ponds . In: Annual Reviews of Fluid Mechanics 14, pp. 153-187.
3. ^ J. Kalff: Limnology: Inland Water Ecosystems. Prentice Hall, New Jersey, 2003. pp. 183-186.
4. ^ A. Horne, C. Goldman: Limnology , 2nd edition. McGraw-Hill, New York et al. A., 1994. pp. 56-60.
5. ↑ ^{a b} S. MacIntyre et al. A. (1999): Boundary mixing and nutrient fluxes in Mono Lake, California . In: Limnology and Oceanography 44 (3), pp. 512-529.
6. ↑ D. Robertson et al. A. (1994): Lake number, a quantitative indicator of mixing used to estimate changes in dissolved oxygen. In: International Review of the Entire Hydrobiology and Hydrography 79 (2), pp. 159–176.