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Some solvable models of shear dispersion

Einige lösbare Scherdispersionsmodelle

Sur quelques modèles resolubles de la dispersion de cisaillement

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Summary

A new method is introduced to solve the turbulent diffusion equation with depth-dependent current. Some simple models are derived from it, which especially discuss the influence of nonlinear current profiles, boundaries and depth-dependence of the exchange coefficients.

Zusammenfassung

Eine neue Methode zur Lösung der turbulenten Diffusionsgleichung mit tiefenabhängiger Geschwindigkeit wird entwickelt. Damit werden einige einfache Modelle aufgestellt, die besonders den Einfluß von nichtlinearen Stromprofilen, undurchdringlichen Wänden und Tiefenabhängigkeit der Austauschkoeffizienten darstellen.

Résumé

On présente une nouvelle méthode pour résoudre l'équation de la diffusion de la turbulence avec un courant dépendant de la profondeur. Quelques modèles simples en dérivent, qui représentent surtout l'influence de profils de courants non linéaires, des limites imperméables et de la dépendance à la profondeur des coefficients d'échange.

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Abbreviations

A nj :

exchange coefficient

A Xν,A Yν A Zν :

element of power series expansion ofA 11 A 22 orA 33

A 1,A 3 :

maximum ofA 11,A 33

A X eff :

effective exchange coefficient (for definition see section 4.2)

a ν :

coefficient of power series expansion of velocityu

b ν :

coefficient of power series expansion of the velocity

C :

calibrated concentration

C N :

approximated calibrated concentration

c ν :

discrete Fourier transform coefficient of the velocity gradient

f :

function (see Appendix 1)

g :

coefficient of the power series expansion off

H :

depth of the channel

h :

half the depth of the channel

i:

\(\sqrt { - 1} \)

j :

index for space co-ordinates (j=1, 2, 3)

K :

maximum of the vertical exchange coefficient (section 4.2)

k, l, m :

indices giving the degree of the moments with respect tox, y, z

M :

total mass

N, N X,N Y,N Z,N α :

boundaries of summation

N BV :

Brunt-Väisälä frequency

n :

index for space co-ordinate (n=1, 2, 3)

P, Q :

dimensionless functions (see section 3.4)

p, q, r :

summation indices (see Appendix 1)

R L :

Lagrangian correlation function

S :

source term

T L(T LX,T LZ):

Lagrangian integral time scale (with respect tox orz)

t :

time

u 0 :

characteristic velocity

u :

velocity component inx direction

u j :

velocity inx j direction

V :

Volume

v :

velocity-component iny direction

W 10 :

wind velocity in 10 m altitude

w :

vertical velocity

w OP :

Okubo-Pritchard diffusion velocity

x :

horizontal co-ordinate (x 1 =x)

x j :

space co-ordinate

y :

horizontal co-ordinate (x 2 =y)

z :

vertical co-ordinate (x 3=z),z positive upwards

α:

power series expansion of the moments with respect to timet

β:

phase

γ:

Hay-Pasquill transformation factor

δ:

Dirac-Delta function

ν, μ, η:

summation indices

ϑ:

arbitrary quantity

σ(σX, σz):

variance (inx, z direction)

σXs :

shear generated part of σX

σXh :

part of σX generated by horizontal turbulence

ω:

angular frequency

{ϑ}:

\(\int\limits_{ - \infty }^\infty {\vartheta CdV} \) ϑCdV weighted mean of ϑ

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  1. Uwe Mikolajewicz

    Present address: , Birkenhof 76, 2390, Flensburg

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    Mikolajewicz, U. Some solvable models of shear dispersion. Deutsche Hydrographische Zeitschrift 39, 1–29 (1986). https://doi.org/10.1007/BF02330520

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    • DOI: https://doi.org/10.1007/BF02330520

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