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Energy- and Flux-Budget Turbulence Closure Model for Stably Stratified Flows. Part II: The Role of Internal Gravity Waves

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Abstract

We advance our prior energy- and flux-budget (EFB) turbulence closure model for stably stratified atmospheric flow and extend it to account for an additional vertical flux of momentum and additional productions of turbulent kinetic energy (TKE), turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). For the stationary, homogeneous regime, the first version of the EFB model disregarding large-scale IGW yielded universal dependencies of the flux Richardson number, turbulent Prandtl number, energy ratios, and normalised vertical fluxes of momentum and heat on the gradient Richardson number, Ri. Due to the large-scale IGW, these dependencies lose their universality. The maximal value of the flux Richardson number (universal constant ≈0.2–0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratification, it increases with increasing wave energy and can even exceed 1. For heterogeneous stratification, when internal gravity waves propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy, reaches zero and then becomes negative. In other words, the vertical flux of potential temperature becomes counter-gradient. Internal gravity waves also reduce the anisotropy of turbulence: in contrast to the mean wind shear, which generates only horizontal TKE, internal gravity waves generate both horizontal and vertical TKE. Internal gravity waves also increase the share of TPE in the turbulent total energy (TTE = TKE + TPE). A well-known effect of internal gravity waves is their direct contribution to the vertical transport of momentum. Depending on the direction (downward or upward), internal gravity waves either strengthen or weaken the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations and large-eddy simulations.

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Abbreviations

A z = E z /E K :

Ratio of the vertical turbulent kinetic energy, E z , to TKE, E K

E = E K + E P :

Total turbulent energy (TTE)

\({E_K =\frac{1}{2}\left\langle {u_i u_i }\right\rangle}\) :

Turbulent kinetic energy (TKE)

E i :

Vertical (i = z) and horizontal (i = x, y) 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) given by Eq. 25

E W :

IGW kinetic energy given by Eq. 16

\({\hat{E}_z}\) :

Dimensionless vertical TKE given by Eq. 70

e :

Vertical unit vector

e W (k):

Energy spectrum of the ensemble of internal gravity waves (IGW) given by Eq. 17

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

Potential-temperature flux

F z :

Vertical component of the potential-temperature flux

\({F_{i}^{W}}\) :

Instantaneous potential-temperature flux caused by the IGW–turbulence interaction given by Eq. 42

\({F_i^{WW}}\) :

Potential-temperature flux caused by IGW averaged over the period of IGW, given by Eq. 21

\({F_\theta ^W}\) :

Scale of the IGW contribution turbulent flux of potential temperature given by Eq. 57

\({\hat{E}_z}\) :

Dimensionless vertical TKE given by Eq. 70

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

Coriolis parameter

G :

“Wave-energy parameter” proportional to the normalized IGW kinetic energy, E W , given by Eq. 50

g :

Acceleration due to gravity

H :

External height scale

K M :

Eddy viscosity given by Eq. 79

K H :

Eddy conductivity given by Eq. 80

k :

Wave vector

k α = (k x , k y ):

Horizontal wave vector with \({k_h =\pm \sqrt{k_x^2 +k_y^2 }}\)

\({k=\sqrt{k_z^2 +k_h^2 }}\) :

Total wavenumber

L :

Obukhov length scale given by Eq. 5

L W :

Minimal wave length of the large-scale IGW

l z :

Vertical turbulent length scale

N :

Mean-flow Brunt–Väisälä frequency

P :

Pressure

P 0 :

Reference value of P

P W :

Pressure variation caused by IGW

p :

Pressure fluctuation caused by turbulence

Pr = ν/κ:

Prandtl number

Pr T :

Turbulent Prandtl number given by Eq. 2

Q :

Dimensionless lapse rate given by Eq. 44

Q ij :

Correlations between the pressure and the velocity-shear fluctuations, given by Eqs. 34 and 63

r :

Radius-vector of the centre of the wave packet

Ri :

Gradient Richardson number, Eq. 1

Ri f :

Flux Richardson number, Eq. 4

\({{Ri}_f^\infty}\) :

Limiting value of Ri f in the flow without IGW (universal constant in the homogeneous sheared flow)

\({{Ri}_f^{\max }}\) :

Limiting value of Ri f in the presence of IGW (depends on the G and Q)

S = |∂U/∂z|:

Vertical shear of the horizontal mean wind

T :

Absolute temperature

T 0 :

Reference value of the absolute temperature

\({t_T =l_z E_z^{-1/2}}\) :

Turbulent dissipation time scale

t τ :

Effective dissipation time scale

\({{\bf V}^{W}=\left({V_1^W, V_2^W, V_3^W}\right)}\) :

IGW velocity given by Eqs. 9–10

\({V_0^W ({\bf k})}\) :

IGW amplitude

U = (U1, U2, U3):

Mean wind velocity

u :

Turbulent velocity

Z 0 :

Height of the IGW source

β = g/T0:

Buoyancy parameter

γ = c p /c v :

Ratio of specific heats

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

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

εα3(eff)(α = 1, 2):

Effective dissipation rates of the vertical turbulent fluxes of momentum

κ :

Temperature conductivity

μ :

Exponent of the energy spectrum of the ensemble of IGW

ν :

Kinematic viscosity

Φ K , Φ θ and Φ F :

Third-order moments representing turbulent fluxes of E K , E θ and F i

φ :

Latitude

ΠW :

IGW production of TKE given by Eq. 49

\({\Pi_i^W}\) :

IGW production of the vertical (i = z) and horizontal (i = x, y) components of TKE given by Eqs. 52–53

\({\Pi_\theta^W}\) :

IGW production of E θ given by Eq. 56

\({\Pi_E^W =\Pi ^{W}+\Pi_\theta ^W \beta ^{2}N^{-2}}\) :

IGW production of TTE

\({\Pi_F^W}\) :

IGW production of the flux of potential temperature given by Eq. 58

\({\Pi_{\tau \alpha }^W}\) :

IGW production of the non-diagonal components of the Reynolds stresses, τ α3, given by Eq. 61

τ ij :

Reynolds stresses characterising the turbulent flux of momentum

τα3(α = 1, 2):

Components of the Reynolds stresses characterising vertical turbulent flux of momentum

τ :

Modulus of (\({\tau_{13}, \tau_{23}}\))

\({\tau_{ij}^W}\) :

Instantaneous Reynolds stresses caused by IGW, given by Eq. 41

\({\tau_{ij}^{WW}}\) :

Reynolds stresses caused by IGW averaged over the period of IGW given by Eqs. 21 and 43

ρ 0 :

Density

Θ:

Potential temperature

Θ W :

IGW potential temperature given by Eq. 11

θ :

Turbulent fluctuation of potential temperature

Ω i :

Earth’s rotation vector parallel to the polar axis

ω :

Frequency of IGW

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Authors and Affiliations

  1. Finnish Meteorological Institute, Helsinki, Finland

    S. S. Zilitinkevich

  2. Division of Atmospheric Sciences, 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

  4. Department of Mechanical Engineering, Pearlstone Centre for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    T. Elperin, N. Kleeorin & I. Rogachevskii

  5. Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel

    V. L’vov

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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. Energy- and Flux-Budget Turbulence Closure Model for Stably Stratified Flows. Part II: The Role of Internal Gravity Waves. Boundary-Layer Meteorol 133, 139–164 (2009). https://doi.org/10.1007/s10546-009-9424-0

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