A brief description of TKESV scheme#
The TKESV turbulence scheme is formulated in terms of two scalar quantities that are approximately conserved for phase changes in the absence of precipitation. These are the total water specific humidity and the liquid water potential temperature. The scheme carries prognostic transport equations (including the time-rate-of-change and the third-order turbulent transport terms) for the turbulence kinetic energy (TKE), the liquid water potential temperature variance, the total water specific humidity variance, and the temperature-humidity covariance. The scalar (co)variances are characteristic of the turbulence potential energy (TPE). The other second-order moments, viz., the Reynolds stress and the fluxes of scalar quantities, are determined through the algebraic diagnostic expressions.
The TKESV scheme utilizes the following parameterization (closure) assumptions. The third-order turbulent transport terms (divergence of the velocity-velocity, pressure-velocity, and velocity-scalar triple correlations) in the TKE, scalar-variance, and scalar-covariance equations are parameterized through the down-gradient approximations. The dissipation terms are parameterized through the relaxation approximations in terms of dissipation time scales. Parameterizations of the pressure-scrambling terms in the Reynolds-stress and scalar-flux equations include the Rotta-type return-to-isotropy formulations of the slow contributions and the linear (in the second-order moments involved) formulations of the rapid contributions. The ensuing “stability functions”, that are known to reveal pathological behaviour over a certain range of their governing parameters (e.g., mean velocity shear and mean buoyancy frequency), are regularized in a physically plausible way. The resulting formulations of the Reynolds stress components and the scalar fluxes are well-behaved over the entire parameter space, provided a stability-dependent formulation of the length/time scale is used. The various time scales, viz., the dissipation time scales in the TKE, scalar-variance, and scalar-covariance equations and the return-to-isotropy time scales in the Reynolds-stress and scalar-flux equations, are set proportional to each other and are expressed in terms of the turbulence length scale and the TKE. An algebraic formulation that accounts for the effect of static stability is used to determine the turbulence length scale. Alternatively, a non-local formulation of the turbulence length scale may be used. A statistical cloud scheme is used to parameterize the effect of latent heat release/consumption on the buoyancy production/destruction of the Reynolds stress and scalar fluxes. A quasi-Gaussian cloud scheme or a scheme that interpolates between the quasi-Gaussian and the exponential formulations is used. The exponential formulation accounts, in a very approximate way, for the skewness of temperature and humidity fields. Lower boundary conditions for the scalar (co)variances account for heterogeneity of the underlying surface and are consistent with the tile approach.
It should be noted that the TKESV scheme is a natural step beyond one-equation TKE schemes, which are (along with more simple algebraic turbulence schemes) still the draft horses of atmospheric turbulence modelling in NWP, climate studies, and related applications. A one-equation TKE scheme is a default option within ICON. The level of complexity of the TKESV scheme is arguably the optimal level of complexity that is required for above applications in terms of essential physics. Since the atmospheric (and oceanic) flows are virtually never hydrostatically neutral, both the TKE and the TPE (characterized by the variances and covariance of scalar quantities) in stratified flows are important. Both energies possess equal rights and should be treated in a similar way. Apart from an improved treatment of the flow energies, the TKESV scheme enables a physically consistent treatment of counter-gradient fluxes of scalars often encountered in geophysical flows (this is not possible with the one-equation TKE schemes) and (potentially) an improved representation of the fractional cloudiness. Furthermore, the TKESV scheme provides a framework for a consistent unified treatment of boundary-layer turbulence and shallow convection.
The code of the TKESV scheme essentially consists of two modules. The main program unit is atm_phy_schemes/mo_turb_tkesv.
It is this module where the actual computations are performed. The common definitions, configuration switches,
and disposable constants and parameters of the TKESV scheme are contained in module atm_phy_schemes/mo_data_tkesv.
The code of the scheme is pretty straightforward. It contains a large number of comment lines
that should help uses to understand the code structure and make changes in the code as needed.
In order to activate the TKESV scheme within ICON, an integer namelist parameter inwp_turb should be set equal to 7.
The TKESV scheme can be somewhat demanding with respect to the time step and may exhibit stability issues at large time steps. Therefore, it is advisable to cap the model time step from above at 120 s. This value is used operationally for ICON global deterministic B3R7N8 runs. Larger time steps may be employed for coarser spatial-resolution configurations, but this is at the user’s own risk.
References
The following publication (in preparation, expected to be published in Reports on ICON series) provides a brief overview of the TKESV scheme, highlights its merits and limitations, and gives details of the implementation of the scheme into ICON. It provides a number of hints regarding the configuration of the TKESV scheme, its tuning, and subtle points that warrant caution.
Mironov, D., E. Machulskaya, and R. Wirth, 2026: Turbulence Kinetic Energy - Scalar Variance turbulence parameterization scheme. Implementation into ICON and some useful hints. (In preparation)
A detailed scientific description of the TKESV scheme is given in
Mironov, D. V., and E. E. Machulskaya, 2017: A turbulence kinetic energy - scalar variance turbulence parameterization scheme. COSMO Technical Report, No. 30, 55 pp. https://doi.org/10.5676/DWD_pub/nwv/cosmo-tr_30
An analysis of the so-called stability functions is given in
Machulskaya, E., and D. V. Mironov, 2020: The stability functions and realizability of the Turbulent Kinetic Energy - Scalar Variance closure for moist atmospheric boundary-layer turbulence. Boundary-Layer Meteorol., 176, 197-228. https://doi.org/10.1007/s10546-020-00528-7
Surface boundary conditions for the scalar (co)variances that are consistent with the tile approach are considered in
Machulskaya, E., and D. Mironov, 2018: Boundary conditions for scalar (co)variances over heterogeneous surfaces. i Boundary-Layer Meteorol., 169, 139-150. https://doi.org/10.1007/s10546-018-0354-6
The implementation of (the earlier version of) the TKESV scheme into a limited-area NWP model COSMO is briefly described in
Machulskaya, E., and D. Mironov, 2013: Implementation of TKE - Scalar Variance mixing scheme into COSMO. COSMO Newsletter, No. 13, 25-33. (available from www.cosmo-model.org)