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A Hierarchy of Energy- and Flux-Budget (EFB) Turbulence Closure Models for Stably-Stratified Geophysical Flows

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Abstract

Here we advance the physical background of the energy- and flux-budget turbulence closures based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth’s rotation. In accordance with modern experimental evidence, the closure implies the maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the two principally different regimes: “strong turbulence” at \({Ri \ll 1}\) typical of boundary-layer flows and characterized by the practically constant turbulent Prandtl number Pr T; and “weak turbulence” at Ri > 1 typical of the free atmosphere or deep ocean, where Pr T asymptotically linearly increases with increasing Ri (which implies very strong suppression of the heat transfer compared to the momentum transfer). For use in different applications, the closure is formulated at different levels of complexity, from the local algebraic model relevant to the steady-state regime of turbulence to a hierarchy of non-local closures including simpler down-gradient models, presented in terms of the eddy viscosity and eddy conductivity, and a general non-gradient model based on prognostic equations for all the basic parameters of turbulence including turbulent fluxes.

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Abbreviations

A i = E i /E K :

Share of the ith-component, E i , of turbulent kinetic energy, E K

E = E K + E P :

Total turbulent energy (TTE)

\({E_{\rm K} = \frac{1}{2}\left\langle {u_i u_i}\right\rangle}\) :

Turbulent kinetic energy (TKE)

E i :

Longitudinal (i = 1 or i = x), transverse (i = 2 or i = y) and vertical (i = 3 or i = z) components of TKE

\({E_\theta = \frac{1}{2}\left\langle {\theta ^{2}}\right\rangle }\) :

“Energy” of potential temperature fluctuations

E P :

Turbulent potential energy (TPE), Eq. 28

\({F_i = \left\langle {u_i \theta}\right\rangle}\) :

Turbulent flux of potential temperature

F z :

Vertical component of F i

\({f = 2\Omega {\rm sin} \varphi}\) :

Coriolis parameter

g :

Acceleration due to gravity

K M :

Eddy viscosity, Eq. 43

K H :

Eddy conductivity, Eq. 44

K D :

Eddy diffusivity

L :

Obukhov length scale, Eq. 41

l :

Turbulent length scale

N :

Mean-flow Brunt–Väisälä frequency

P :

Mean pressure

P 0 :

Reference value of P

p :

Fluctuation of pressure

Pr = ν/κ :

Prandtl number

Pr T :

Turbulent Prandtl number, Eq. 45

Q ij :

Correlations between fluctuations of pressure and velocity shear, Eq. 15

Ri :

Gradient Richardson number, Eq. 3

Ri f :

Flux Richardson number, Eq. 40

R ∞ :

Maximal Ri in homogeneous sheared flow

S = |∂ U/∂ z|:

Vertical shear of the horizontal mean wind

T :

Absolute temperature

T 0 :

Reference value of absolute temperature

\({t_{\rm T} = l E_{\rm K}^{-1/2}}\) :

Dissipation time scale

t τ :

Effective dissipation time scale

U = (U 1, U 2, U 3):

Mean wind velocity

u = (u 1, u 2, u 3):

Velocity fluctuation

β = g/T 0 :

Buoyancy parameter

γ = c p /c v :

Ratio of specific heats at constant pressure and constant volume

\({\varepsilon_{\rm K}, \varepsilon_\theta, \varepsilon_i^{\rm (F)}}\) and \({\varepsilon_{ij}^{(\tau )}}\) :

Dissipation rates for \({E_{\rm K}, E_\theta, F_i^{\rm (F)}}\) and τ ij

\({\varepsilon_{\alpha 3({\rm eff})}^{(\tau)}}\) (α = 1, 2):

Effective dissipation rates for the vertical turbulent fluxes of momentum

κ :

Temperature conductivity

ν :

Kinematic viscosity

Π = E P/E K :

Energy stratification parameter, Eq. 74

ΦK, Φ θ and ΦF :

Third-order turbulent fluxes of TKE E K, and the fluxes of E θ and F i

\({\varphi}\) :

Latitude

τ ij :

Reynolds stresses (components of turbulent flux of momentum)

τ α3 (α = 1, 2):

Components of the Reynolds stresses representing the vertical turbulent flux of momentum

τ :

Modulus of the horizontal vector (τ13, τ 23)

ρ :

Mean density

ρ 0 :

Reference value of ρ

Θ:

Mean potential temperature

θ :

Fluctuation of potential temperature

Ω:

Angular velocity of Earth’s rotation

Ω i :

Earth’s rotation vector (parallel to the polar axis)

C 0 = 0.125:

Inter-component energy exchange constant determining the vertical share of TKE, Eqs. 49, 50c

C 1 = 0.5, C 2 = 0.72:

Inter-component energy exchange constants determining the longitudinal and transverse shares of TKE, Eqs. 48– 50

C F = 0.25:

Dissipation time scale constant for the turbulent flux of potential temperature, Eq. 19

C P = 0.86:

Dissipation time-scale constant for the turbulent flux of TPE, Eq. 19

C r = 1.5:

Standard inter-component energy exchange constant, Eqs. 27, 50a, 50b, 50c

C τ = 0.2:

Dissipation time-scale constant for the turbulent flux of momentum, Eq. 33

C Ω = 1:

Rotational length-scale constant, Eq. 73

R ∞ = 0.25:

Upper limit for the flux Richardson number attainable in the steady- state regime of turbulence, Eqs. 40, 46

k = 0.4:

von Karman constant, Eq. 67

a 1 = 0.18, a 2 = 0.16,:

a 3 = 1.42 in Eqs. 81– 86

C u = k/R ∞ = 1.6:

In the velocity gradient formulation, Eq. 70

C θ = 0.105:

In Eqs. 36, 37, 47, 64

k T = (C F/C τ )k = 0.5:

von Karman constant for temperature, Eq. 86

\({Pr_{\rm T}^{(0)} = 0.8}\) :

Turbulent Prandtl number in neutral stratification, Eq. 57

Π∞ = 0.14:

Upper limit for the energy stratification parameter, Eq. 76

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Authors and Affiliations

  1. Finnish Meteorological Institute, Helsinki, Finland

    S. S. Zilitinkevich

  2. Division of Atmospheric Sciences and Geophysics, University of Helsinki, Helsinki, Finland

    S. S. Zilitinkevich

  3. Nansen Environmental and Remote Sensing Centre, Bjerknes Centre for Climate Research, Bergen, Norway

    S. S. Zilitinkevich & I. Esau

  4. Department of Radio Physics, N.I. Lobachevski State University of Nizhniy Novgorod, Nizhny Novgorod, Oblast, Russia

    S. S. Zilitinkevich

  5. A.M. Obukhov Institute of Atmospheric Physics, Moscow, Russia

    S. S. Zilitinkevich

  6. Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    T. Elperin, N. Kleeorin & I. Rogachevskii

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  1. S. S. Zilitinkevich
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  2. T. Elperin
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Zilitinkevich, S.S., Elperin, T., Kleeorin, N. et al. A Hierarchy of Energy- and Flux-Budget (EFB) Turbulence Closure Models for Stably-Stratified Geophysical Flows. Boundary-Layer Meteorol 146, 341–373 (2013). https://doi.org/10.1007/s10546-012-9768-8

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