INTRODUCTION

Exchange processes at the water–air interface are the key factor of hydrodynamic and environmental processes in aquatic ecosystems, the formation of weather and climate, the origination of currents, surface waves, and turbulent mixing, which has a direct effect on the transport of solutes [51], the characteristics of stratification [64], the oxygen regime and gas exchange [38]. The main characteristics of the interaction are vertical turbulent fluxes of momentum, heat, and moisture (sensible and latent heat). The knowledge of the magnitudes of these fluxes is necessary for numerical weather forecasts, simulation of the Earth system, interpretation of remote sensing data, and other geophysical applications.

The lack of knowledge on the structure of the surface layer of the atmosphere and its exchange of momentum, heat, and moisture with a rough water surface under different background conditions is now the main obstacle for the correct functioning of operational, global, and regional weather forecast models and expert models of climate change.

The transport of momentum between the atmosphere and water surface is largely determined by the roughness characteristics of momentum, temperature, and humidity. In addition, the momentum flux is affected by wind speed; atmospheric stratification; and the size, steepness, and phase velocity of wind waves and swell. The issue of the properties of the surface roughness parameters of shallow water areas is still largely open, despite numerous studies. This is especially true for small water areas, where wind wave parameters do not depend on the fetch and are determined by wind direction and topography characteristics.

In the numerical simulation of the boundary layer, the averaged fluxes are calculated with the use of the so-called aerodynamic bulk formulas [24]:

$$\tau = \rho {{C}_{D}}U_{z}^{2},$$
(1)
$$H = \rho {{c}_{p}}{{C}_{H}}{{U}_{z}}\left( {{{T}_{s}} - {{T}_{z}}} \right),$$
(2)
$$LH = {{L}_{s}}\rho {{C}_{E}}{{U}_{z}}\left( {{{q}_{s}} - {{q}_{z}}} \right),$$
(3)

where CD, CH, CE are dimensionless exchange coefficients (the resistance factor, Stanton number, and Dalton number, respectively); cp and ρ are air heat capacity and density, Ls is boiling heat; τ, H, and LH are turbulent fluxes of momentum, heat, and moisture, respectively; Uz, Tz, and qz are wind speed and air temperature and humidity at elevation z, respectively; Ts and qs are the temperature and humidity at the surface. The exchange coefficients commonly refer to a standard elevation of measurements z = 10 m and to neutral stratification conditions.

This method enables the use of data of standard meteorological measurements; however, the main problem is the evaluation of exchange coefficients.

The equations of the Monin–Obukhov similarity theory (MOST) [4] yield the expressions:

$${{C}_{D}} = \frac{\kappa }{{\left[ {\ln \frac{z}{{{{z}_{{0u}}}}} - {{\Psi }_{m}}\left( {\frac{z}{L}} \right) + {{\Psi }_{m}}\left( {\frac{{{{z}_{{0u}}}}}{L}} \right)} \right]}},$$
(4)
$${{C}_{H}} = {{\alpha }_{T}}{{C}_{D}}\frac{{\left[ {\ln {z \mathord{\left/ {\vphantom {z {{{z}_{{ou}}}}}} \right. \kern-0em} {{{z}_{{ou}}}}} - {{\Psi }_{u}}\left( {{z \mathord{\left/ {\vphantom {z L}} \right. \kern-0em} L}} \right) + {{\Psi }_{u}}\left( {{{{{z}_{{0u}}}} \mathord{\left/ {\vphantom {{{{z}_{{0u}}}} L}} \right. \kern-0em} L}} \right)} \right]}}{{\left[ {\ln {z \mathord{\left/ {\vphantom {z {{{z}_{{oT}}}}}} \right. \kern-0em} {{{z}_{{oT}}}}} - {{\Psi }_{T}}\left( {{z \mathord{\left/ {\vphantom {z L}} \right. \kern-0em} L}} \right) + {{\Psi }_{T}}\left( {{{{{z}_{{0T}}}} \mathord{\left/ {\vphantom {{{{z}_{{0T}}}} L}} \right. \kern-0em} L}} \right)} \right]}},$$
(5)
$${{C}_{E}} = {{\alpha }_{q}}{{C}_{D}}\frac{{\left[ {\ln {z \mathord{\left/ {\vphantom {z {{{z}_{{ou}}}}}} \right. \kern-0em} {{{z}_{{ou}}}}} - {{\Psi }_{u}}\left( {{z \mathord{\left/ {\vphantom {z L}} \right. \kern-0em} L}} \right) + {{\Psi }_{u}}\left( {{{{{z}_{{0u}}}} \mathord{\left/ {\vphantom {{{{z}_{{0u}}}} L}} \right. \kern-0em} L}} \right)} \right]}}{{\left[ {\ln {z \mathord{\left/ {\vphantom {z {{{z}_{{oq}}}}}} \right. \kern-0em} {{{z}_{{oq}}}}} - {{\Psi }_{T}}\left( {{z \mathord{\left/ {\vphantom {z L}} \right. \kern-0em} L}} \right) + {{\Psi }_{q}}\left( {{{{{z}_{{0q}}}} \mathord{\left/ {\vphantom {{{{z}_{{0q}}}} L}} \right. \kern-0em} L}} \right)} \right]}},$$
(6)

where \({{\alpha }_{T}} = {{{{K}_{T}}} \mathord{\left/ {\vphantom {{{{K}_{T}}} {{{K}_{m}}}}} \right. \kern-0em} {{{K}_{m}}}}\) and \({{\alpha }_{q}} = {{{{K}_{q}}} \mathord{\left/ {\vphantom {{{{K}_{q}}} {{{K}_{m}}}}} \right. \kern-0em} {{{K}_{m}}}}\) are the ratios of turbulent thermal conductivity and diffusion coefficients to viscosity, or inverse turbulent Prandtl and Schmidt numbers, respectively; z is measurement height; \({{z}_{{0u}}}\) is the parameter of dynamic (or aerodynamic) roughness; \({{z}_{{0T}}}\) and \({{z}_{{0q}}}\) are the roughness parameters for temperature and specific humidity, i.e., elevations at which the wind speed, temperature, and humidity reach surface values, if the profile of appropriate meteorological characteristics is extrapolated to the surface. The integral universal functions \({{\Psi }_{a}}\) are defined as follows: \({{\Psi }_{a}} = \int_0^\zeta {\frac{{1 - {{\phi }_{a}}(\zeta )}}{\zeta }} d\zeta \), \({{\phi }_{a}}(\zeta )\) are universal functions, describing the profiles of meteorological characteristics, \(\zeta = \frac{z}{L}\) is a dimensionless stability parameter, \(L = \frac{{u_{*}^{3}}}{{\kappa \left( {\frac{{\text{g}}}{{{{T}_{0}}}}} \right)\left( {\frac{H}{{{{c}_{p}}\rho }}} \right)}}\) is Obukhov’s scale.

The parameterizations for determining the universal functions have been developed based on numerous special experiments under different stratification conditions [16, 23, 32, 53] and adequately describe the profiles of meteorological characteristics under the conditions of stationarity and homogeneous relief. The roughness characteristics \({{z}_{{0u}}}\), \({{z}_{{0T}}}\), and \({{z}_{{0q}}}\) are physical variables not measured directly. They are introduced into similarity formulas in order to avoid detailed description of wind speed and temperature profiles in the immediate vicinity to the underlying surface, i.e., in the viscous sublayer. In MOST, the roughness parameters determine the interaction between the viscous sublayer, in which momentum and scalar variables are transferred by molecular heat conductance, diffusion, and viscosity, and the rough surface [14, 30]. In the viscous sublayer, the transport of momentum through the surface is mainly due to the pressure difference on opposite faces of the roughness elements, and the heat transfer is due to molecular thermal conductivity. This contrast leads to a difference between the roughness scales for wind speed \({{z}_{{0u}}}\) (dynamic roughness) and for scalars (in particular, temperature and humidity) [79]. The parameter of dynamic roughness for different land surfaces in a developed turbulent flow is primarily determined by the sizes and shapes of roughness elements [14, 30] (except for the cases where the height of roughness elements is comparable with the Obukhov length scale, for example, in the case of urban development [80]), i.e., it can be evaluated based on the structure of the surface. The roughness parameters for the temperature and humidity are more variable and depend on a larger number of factors, including molecular viscosity and heat conductance [6, 66].

In the case of sea surface, the evaluation of dynamic roughness parameter is complicated by the dependence of the state of the surface on wind speed. Notwithstanding the numerous works in this field, there is still no clear understanding of the character of the dependence of the dynamic roughness parameter and the resistance of the water surface, associated with it, on the mean wind speed, wave character and age, and the dynamic and temperature state of the sea surface. The roughness of water surface is due to the motion of the air layer in contact with it and is maintained mostly by the transfer of momentum and energy to the surface. The small-scale roughness of the sea surface is a complex combination of gravity waves and capillary ripples, the origin and structure of which depend not only on the wind, but also on currents, internal waves in the sea and atmosphere, water body depth, bed topography, the effect of moving and stationary objects, anthropogenic surface pollution, and other factors [21, 25].

In terms of the impact of water bodies on the climate system, of particular interest are inland water bodies, i.e., lakes and reservoirs [19, 56]. The surface of lakes differs from the surrounding landscape by a small albedo, much lesser roughness, high thermal conductance, and high heat capacity; therefore, they have an effect on the processes in the atmospheric boundary layer, local atmospheric circulation, and the heat and water balance at regional scale [31, 46], which should be taken into account in the regional and climatic simulation.

However, in the majority of lake models, the schemes of parameterization of exchange processes are still based on oceanic data [33, 47, 68, 79]. Because of the difference between wave formation processes in an ocean and a lake, associated with the depth of the water body and the limited wave run, this approach can lead to significant errors [62]. Thus, it was found that lake surface can be aerodynamically rougher than the open ocean, the wind speed being the same; the use of oceanic parameterizations may lead to an error in the annual estimate of evaporation over the lake by 40% [44]. Therefore, attention should be paid to the parameterization of the exchange coefficients and roughness characteristics for lake models. Studies in this direction have been carried out before [13, 34, 35, 52, 67], but the issue of the properties of the roughness parameter for the surface of shallow water areas, in particular, lakes, is still largely open.

The main criterion for determining wind waves in shallow areas is \(H \leqslant {\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0em} 2}\) (H is water body depth, and λ is the characteristic wave length) [2]. For the conditions of deep water, the largest resistance to the wind is due to the high-frequency components of the spectrum of sea waves; because their phase velocities are much less than the phase velocities in the vicinity of the spectral maximum of waves \({{\omega }_{0}}\), and hence, the air flow velocity. The long and gently sloping waves, corresponding to the maximum of the spectrum of sea waves and having phase velocities close to the wind speed, do not produce appreciable tangential resistance to air flow, but can introduce wave resistance. The waves in shallow areas have relatively low phase velocities because of the limiting effect of water depth at a relatively high steepness, which is due to the nonlinear interaction between long and short waves. Because of this, the contribution of the components of wave spectrum to the total resistance of water surface near the maximum of the spectrum with a frequency \({{\omega }_{0}}\) is comparable with the contribution of the high-frequency components and even predominant. This effect increases with a decrease in water body depth, i.e., the parameter of dynamic roughness increases with increasing wind speed or decreasing depth. In addition, asynchronous interaction between changes in surface wave characteristics and wind speed field takes place in small lakes. An important distinction of wave formation in lakes compared with that in the ocean is the short fetch. As the result, the wave field is characterized by young and high-frequency waves, and the measured values of wave age exceed the literature data. Therefore, the estimates obtained in the open ocean and even in shallow coastal zone are inapplicable to the parameterization of exchange processes in inland water bodies.

A fact that is also not taken into account is that the conditions of gentle winds are more typical of inland water bodies than of an open ocean [77], and in this case, the roughness characteristics show effects of the heterogeneity of the surface tension and small-scale capillary waves [48, 75].

Taking into account that the sensitivity of the determination of turbulent fluxes and the atmosphere–water surface interface to the choice of the scheme for determining roughness parameters is high [74], it is necessary to develop reliable schemes for calculating these parameters, in particular, in the case of shallow water bodies.

METHODS FOR DETERMINING THE ROUGHNESS PARAMETER

According to MOST, the roughness parameters \({{z}_{{0u}}}\), \({{z}_{{0T}}}\), and \({{z}_{{0q}}}\) are determined as elevations at which the profiles of the appropriate weather characteristics are zero.

$$u = \frac{{{{u}_{*}}}}{\kappa }\left[ {\ln \left( {\frac{z}{{{{z}_{{0u}}}}}} \right) - {{\Psi }_{u}}\left( {\frac{z}{L}} \right) + {{\Psi }_{u}}\left( {\frac{{{{z}_{{0u}}}}}{L}} \right)} \right].$$
(7)

For the temperature (and in a similar way, for humidity)

$$T = {{T}_{s}} + \frac{{{{T}_{*}}}}{{{{\kappa }_{T}}}}\left[ {\ln \left( {\frac{z}{{{{z}_{{0T}}}}}} \right) - {{\Psi }_{T}}\left( {\frac{z}{L}} \right) + {{\Psi }_{T}}\left( {\frac{{{{z}_{{0T}}}}}{L}} \right)} \right],$$
(8)

\({{T}_{*}} = \frac{H}{{{{u}_{*}}}}\), Ts is the temperature of the surface or the aerodynamic temperature (air temperature at roughness height).

Formulas (4)–(6) and (7), (8), yield the following relationships for the roughness parameters

$${{z}_{{0u}}} = z\exp \left\{ { - \left[ {\frac{\kappa }{{\sqrt {{{C}_{D}}} }} + {{\Psi }_{m}}\left( {\frac{z}{L}} \right) - {{\Psi }_{m}}\left( {\frac{{{{z}_{0}}}}{L}} \right)} \right]} \right\},$$
(9)
$${{z}_{{0T}}} = z\exp \left\{ { - \left[ {\frac{{\kappa \sqrt {{{C}_{D}}} }}{{{{C}_{H}}}} + {{\Psi }_{h}}\left( {\frac{z}{L}} \right) - {{\Psi }_{h}}\left( {\frac{{{{z}_{0}}}}{L}} \right)} \right]} \right\},$$
(10)
$${{z}_{{0q}}} = z\exp \left\{ { - \left[ {\frac{{\kappa \sqrt {{{C}_{D}}} }}{{{{C}_{E}}}} + {{\Psi }_{h}}\left( {\frac{z}{L}} \right) - {{\Psi }_{h}}\left( {\frac{{{{z}_{0}}}}{L}} \right)} \right]} \right\}.$$
(11)

In this study, universal functions are used in the following form [16, 23, 27, 32, 36]:

unstable stratification \(\zeta < - 0.05\):

$${{\Psi }_{u}} = \frac{{{{\Psi }_{{{\text{kanzas}}}}} + {{\zeta }^{2}}{{\Psi }_{{{\text{conv}}}}}}}{{1 + {{\zeta }^{2}}}},$$
(12)
$$\begin{gathered} {{\Psi }_{{{\text{kanzas}}}}} = 2\ln \left( {\frac{{1 + x}}{2}} \right) \\ + \,\,\ln \left( {\frac{{1 + {{x}^{2}}}}{2}} \right) - 2\arctan x + \frac{\pi }{2}, \\ \end{gathered} $$
(13)
$$\begin{gathered} {{\Psi }_{{{\text{conv}}}}} = \frac{3}{2}\ln \left( {\frac{{{{y}^{2}} + y + 1}}{3}} \right) - \sqrt 3 \arctan \frac{{2y + 1}}{{\sqrt 3 }} + \frac{\pi }{{\sqrt 3 }}, \\ x = {{\left( {1 - 19.3~\zeta } \right)}^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0em} 4}}}},\,\,\,\,y = {{\left( {1 - 13\zeta } \right)}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0em} 3}}}}, \\ \end{gathered} $$
(14)
$$\begin{gathered} {{\Psi }_{T}}(\zeta ) = 2\ln \left( {\frac{{1 + {{x}^{2}}}}{2}} \right), \\ x = 0.95{{\left( {1 - 11.6\zeta } \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}; \\ \end{gathered} $$
(15)

neutral stratification: \( - 0.05 < \zeta < 0.05\):

$${{\Psi }_{u}}\left( \zeta \right) = 0,\,\,\,\,C_{{Dn}}^{{}} = C_{D}^{{}};$$
(16)

stable stratification \(\zeta > 0.05\):

$${{\Psi }_{u}}(\zeta ) = - 6\zeta {\text{,}}\,\,\,\,{{\Psi }_{T}}(\zeta ) = - 7.8\zeta {\kern 1pt} .$$
(17)

The determination of the coefficients of exchange and Obukhov scale (stability parameter) by this method requires direct pulsation measurements of atmospheric turbulence characteristics. Despite the fact that the calculation uses empirical universal functions, this method is the only direct way to determine the roughness parameter.

When profile measurements are available, the dynamic roughness parameter can be calculated with the use of a formula for determining logarithmic wind profile under neutral stratification \(u = \frac{{{{u}_{*}}}}{\kappa }\left[ {\ln \left( {\frac{z}{{{{z}_{{0u}}}}}} \right)} \right]\). In this case, \({{z}_{{0u}}}\) can be readily determined by measurements at two levels

$$\ln {{z}_{{0u}}} = \frac{{\ln {{z}_{2}} - \frac{{{{u}_{2}}}}{{{{u}_{1}}}}\ln {{z}_{1}}}}{{1 - \frac{{{{u}_{2}}}}{{{{u}_{1}}}}}},$$
(18)

\({{z}_{1}}\) and \({{z}_{2}}\) are the upper and lower levels at which wind speeds \({{u}_{1}}\) and \({{u}_{2}}\) are measured.

Taking into account that, at the interaction between the atmosphere and the rough surface, there exists a displacement height D, to which the atmospheric flow shifts in the vertical direction [80], the formulas for the logarithmic profile of wind and (18) are to be rewritten as

$$\frac{{{{U}_{z}}}}{{{{u}_{*}}}} = {{\kappa }^{{ - 2}}}\ln \left( {\frac{{z - D}}{{{{z}_{{0u}}}}}} \right),$$
(19)
$$\begin{gathered} {{z}_{{0u}}} = \left( {z - D} \right)\exp \left( {\frac{{ - \kappa {{U}_{z}}}}{{{{u}_{*}}}}} \right) \\ = {{\left( {{{z}_{2}} - {{z}_{1}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{z}_{2}} - {{z}_{1}}} \right)} {\left[ {\exp \left( {\frac{{ - \kappa {{u}_{2}}}}{{{{u}_{*}}}}} \right) - \left( {\frac{{ - \kappa {{u}_{1}}}}{{{{u}_{*}}}}} \right)} \right]}}} \right. \kern-0em} {\left[ {\exp \left( {\frac{{ - \kappa {{u}_{2}}}}{{{{u}_{*}}}}} \right) - \left( {\frac{{ - \kappa {{u}_{1}}}}{{{{u}_{*}}}}} \right)} \right]}}, \\ D = {{z}_{1}} - \frac{{{{z}_{2}} - {{z}_{1}}}}{{\exp \left( {\frac{{ - \kappa ({{u}_{2}} - {{u}_{1}})}}{{{{u}_{*}}}}} \right) - 1}}. \\ \end{gathered} $$
(20)

Formula (20) requires additional data on the dynamic velocity. However, the displacement depth D of the water surface is insignificant, especially, in the case of weak and moderate winds, therefore, it can be ignored.

The calculation of the roughness parameter from (18) under real conditions can lead to errors because of the deviation of the actual wind speed profile from the logarithmic law at atmospheric stratification other than neutral. The application of the profile method is justified if the measurement levels of meteorological parameters are not high; it is applicable only for fixed basements and requires wind speed measurements by the same type of sensors with a high accuracy. The values of z0u thus obtained from the logarithmic profile, constructed by data of u(z) at two levels, differ only slightly even in the cases of highly stable and highly unstable stratification, if the measurements are made at levels of up to ~5 m above the surface [42]. Corrections for stratification due to the functions \({{\Psi }_{u}}\left( \zeta \right)\) and \({{\Psi }_{T}}\left( \zeta \right)\) are also small at strong winds, when \(L \to \pm \infty \) and \({{\Psi }_{u}}\left( \zeta \right)\), \({{\Psi }_{T}}\left( \zeta \right) \to 0\).

In the recent decades, various schemes were developed for the evaluation of the roughness parameter of water surfaces [25, 45, 57, 68], some of which take into account the stratification of the atmosphere [79].

The parameterization of the roughness conditions at the water–air interface in the model of interaction between the atmosphere and the ocean is commonly made with the use of Charnock formula [18]:

$${{z}_{{0u}}} = a\frac{{u_{*}^{2}}}{{\text{g}}},$$
(21)

where a is an empirical coefficient referred to as Charnock parameter. Experimental data [12] show that the values of Charnock coefficient can vary within more than an order of magnitude, depending on the conditions, and depend significantly on the characteristics of surface waves and water body depth. Various attempts were made to relate the roughness parameter with the parameters of sea waves. Based on numerous laboratory and field experiments and theoretical calculations, the parameter characterizing the roughness of the sea surface was taken to be the age of waves, defined as \(\left( {\frac{{{{c}_{{ph}}}}}{{{{u}_{*}}}}} \right)\) or \(\left( {\frac{{{{c}_{{ph}}}}}{{{{u}_{z}}}}} \right)\), where cph is wave phase velocity. Additional parameters are the frequency of the spectral peak of wind waves [63] and wave height.

Some researchers [22, 26, 39, 40, 49, 58] proposed a generalized formula to describe the dependence of the roughness coefficient on the wave age parameter in the form

$$\frac{{{\text{g}}{{z}_{{0u}}}}}{{u_{*}^{2}}} = f\left( {\frac{{{{c}_{{ph}}}}}{{{{u}_{*}}}}} \right).$$
(22)

For the limiting development of waves in shallow water bodies, when the surface roughness characteristics cease to depend on the fetch, as well as in the case of the transformation of waves coming from the open sea (large fetches), the following equation is used [5]

$$\frac{{{\text{g}}{{z}_{{0u}}}}}{{u_{*}^{2}}} = f\left( {\frac{{{\text{g}}h}}{{u_{*}^{2}}}} \right),$$
(23)

where h is water body depth. Based on dimensional considerations, the dependence (23) can be approximated by the formula

$${{z}_{{0m}}} = m\frac{{u_{*}^{4}}}{{{{{\text{g}}}^{2}}h}}.$$
(24)

The value of the coefficient m is chosen using the conditions that in the case of developed waves and deep sea, Charnock parameter \(a = \frac{{{\text{g}}{{z}_{{0u}}}}}{{u_{*}^{2}}} \to 0.011\); it varies from 25 to 50, depending on the depth of the water body and the distance from the shore. The measurement data show that this approach is applicable to coastal zones, but in some cases, it cannot be used for closed inland water bodies.

In the case of an oceanic surface, as Charnock supposed in his theoretical studies, the parameter a has an order of magnitude of 10−2 ([17, 18], a = 0.0123). This was confirmed by later studies (in [57], a = 0.011). According to [7], the values of Charnock parameter for shallow water bodies can differ considerably from those for the ocean. For example, higher values of a were found in studies [29] (a = 0.0144), [76] (a = 0.018), and [55] (greater by an order of magnitude: а = 0.110). For lakes and coastal zones, some researchers substantiate the value а = 0.03 [12, 29, 59, 70].

Under light winds, the water surface can be regarded as a smooth solid wall, separated from the near-wall flow by a viscous sublayer. The predominance of the viscous mechanism of roughness parameter formation of sea surface is observed at wind speed <5 m/s. Therefore, in the schemes of aerodynamic roughness parameterization, two situations are commonly considered: a current associated with wind stress [18, 57] and a current associated with viscosity.

With viscosity taken into account, the Charnock formula can be rewritten as

$${{z}_{{ou}}} = c\frac{\nu }{{{{u}_{*}}}} + a\frac{{u_{*}^{2}}}{{\text{g}}},$$
(25)

where ν is air kinematic viscosity (m2/s), which is equal to ⁓1.5 × 10–5 m2/s for the atmosphere at sea level. The parameter c is related to the Reynolds number Re, and, under oceanic conditions, it is taken equal to 0.11; in the case of lakes it can increase to 0.54 [72].

The roughness parameters for temperature and humidity are evaluated with the use of an approach first proposed by S.S. Zilitinkevich [1], where its dependence on Re and the parameter of dynamic roughness is parameterized. In the general form, this relationship can be written as

$$\ln \frac{{{{z}_{{0u}}}}}{{{{z}_{{0T,q}}}}} = a{{\operatorname{Re} }^{n}} + b,\,\,\,\,\operatorname{Re} = \frac{{{{z}_{{0u}}}{{u}_{*}}}}{\nu }.$$
(26)

The coefficients a and b depend on the type of the surface and are determined from measurement data; the exponent n corresponds to various simplifications in equations of transport–diffusion of a scalar variable (temperature, humidity) in a viscous sublayer [43]. In different parameterizations, the values n = 1; 0.5; 0.25 are used [66]. Formula (26) can be rewritten as [62]:

$$\ln \frac{{{{z}_{{0u}}}}}{{{{z}_{{0T}}}}} = \frac{\kappa }{{\Pr }}(a{{\operatorname{Re} }^{n}}\, + b),\,\,\,\,\ln \frac{{{{z}_{{0u}}}}}{{{{z}_{{0q}}}}} = \frac{\kappa }{{{\text{Sc}}}}(a{{\operatorname{Re} }^{n}}\, + b),$$
(27)

where \(\kappa \) is Karman’s constant, Pr =  0.71 is molecular Prandtl number for air, Sc =  0.66 is molecular Schmidt number of water vapor.

Reviews of roughness parameterizations for various surfaces can be found in [6, 14]. In the case of sea surface, the parameterization proposed in S.S. Zilintkevich study [79] is used. For the case of lakes, roughness parameterizations were tested in [20, 59, 62, 67, 71]. However, lake models, as before, are mostly based on the parameterizations developed for the open sea.

DATA AND PROCESSING METHODS USED

The parameterizations of the roughness parameters are analyzed using data of measurements of atmospheric turbulence characteristics in Bol’shoi Vilyui Lake [61] and the Mozhaisk and Gorky reservoirs. The average depth of Bol’shoi Vilyui Lake is 4 m, and that of the Mozhaisk and Gorky reservoirs is 20 m. On the lake and the Mozhaisk Reservoir, the measurement complex was located at a distance from the shores on an anchored floating base. In the Gorky Reservoir, Geofizik catamaran was used for measurements [3]. During the measurements, the direction of wind ensured a sufficient length of wave fetch, thus allowing the effect of shore to be neglected. The roughness parameters were determined by formulas (9)–(11), where the values of the coefficients of exchange, dynamic velocity \({{u}_{*}}\), heat and moisture fluxes, and the stability parameter \(\zeta = \frac{z}{L}\) where used. In formulas (24) and (25), the values of \({{u}_{*}}\) where also taken from measurements. The measurements were carried out on the Bol’shoi Vilyui in the summer of 2015; on the Mozhaisk Reservoir, in 2017; and on the Gorky Reservoir, in 2016–2018.

The measurement system consisted of an acoustic anemometer (WindMaster 3D brand, manufactured by Gill Instruments) and an open infrared CO2/H2O gas analyzer (LI 7500 brand, manufactured by LI-COR, Inc.). The gas analyzer was not used in Bol’shoi Vilyui Lake, where only the dynamic roughness parameter and the roughness parameter for temperature were determined. The data of acoustic anemometer were synchronized with gas analyzer data and used to calculate turbulent heat, momentum, and methane fluxes. The fluxes were calculated with the use of turbulent pulsation method (Eddy-covariance) [15].

In this method, the fluxes are calculated by covariations between the measured pulsations of meteorological parameters and gas concentrations:

$$\tau = {{\rho }_{0}}u_{*}^{2} = - {{\rho }_{0}}\left[ {{\mathbf{i}}\overline {u{\kern 1pt} 'w{\kern 1pt} '} + {\mathbf{j}}\overline {{v}{\kern 1pt} 'w{\kern 1pt} '} } \right],$$
(28)
$$H = {{c}_{p}}{{\rho }_{0}}\overline {w{\kern 1pt} 'T{\kern 1pt} '} ,$$
(29)
$$LH = {{L}_{s}}\rho {{C}_{E}}\overline {w{\kern 1pt} 'q{\kern 1pt} '} .$$
(30)

The denotations correspond to those in (1)–(3); u', v', w' are pulsations of the three components of wind velocity: longitudinal, transverse, and vertical, respectively; T ' is temperature pulsation; q' is humidity pulsation. The calculation of fluxes was made with the use of spectral correction [50], density fluctuation correction [73], acoustic temperature correction [65], anemometer slope correction, as well as statistical tests [69]. Data quality control was carried out with the use of methods proposed in [28]. The footprint (the zone of flow formation on the surface) was evaluated using an analytical model [41]. The averaging period was taken equal to 20 min.

However, even at all corrections, the data on turbulent fluxes commonly have a large random spread. Accordingly, the roughness parameters evaluated by formulas (9)–(11) also show a wide spread. The only way to cope with this spread is to use additional corrections [9–11, 78]. The selection of data was carried out with the use of limitations on wind speed \({{u}_{*}} > 0.05\) m/s and on the values of fluxes \(\left| {H,LH} \right| \geqslant 10\) W/m2. Only the data corresponding to the domain of flow formation over water surface were used. During additional verification, the data were rejected if they met the following criteria [44]:

$${{z}_{{0u}}},{{z}_{{0T}}},{{z}_{{0q}}} \geqslant 0.3\,\,{\text{m}},$$
(31)
$${{z}_{{0u}}} \leqslant {{10}^{{ - 8}}}\,{\text{m}}{\text{,}}$$
(32)
$${{z}_{{0T}}},{{z}_{{0q}}} \leqslant {{10}^{{ - 7}}}\,{\text{m}}{\text{.}}$$
(33)

The criterion (31) was applied in accordance with the assumption that the roughness scale does not exceed one tenth of the observation point elevation (maximum 3.0 m in this study). The criteria (32), (33) were used taking into account the transport of heat and moisture in the viscous sublayer [60]. It is supposed that at scales less than this level, the surface exchange of heat and moisture cannot take place [8, 11].

As the result of filtrations, 645 values of dynamic roughness parameter, 572 values of roughness parameter for the temperature, and 489 values for humidity were selected for the analysis.

MEASUREMENT RESULTS

Figure 1 shows the Charnock parameter obtained from the measurements (formulas (9), (21)). In the case of wind speed >5 m/s, the Charnock parameter is equal to 0.03, which is in agreement with the earlier data [71]. The measurements confirm the predominance of the viscous mechanism of roughness formation at winds <4 m/s, when the calculation by formula (21) is incorrect, and the effects of viscosity are to be taken into account. The wide spread of the experimental data on the roughness parameter at gentle winds can be due to the stratification, nonstationary wind field, the effect of direction, and low speed of wind [54].

Fig. 1.
figure 1

Dependence of Charnock parameter (21), obtained from measurements in shallow-water areas, on wind speed U.

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Figure 2 compares the parameters of dynamic roughness, obtained by formulas (9), (25) (c = 0.11, a = 0.03), and (24) (m = 25). At medium wind speeds, the calculations by Charnock formula coincide with observation results. The overestimates by (24) are likely due to the fact that the parameterization was carried out for coastal regions with transformation of waves coming from the open sea. The values of the roughness parameter in the absence of wind \({{z}_{0}} = 0.001\) m correspond to the data of model calculations for free convection regime in [37]. Overall, the parameter of dynamic roughness varies from 0.00007 to 0.0009 m.

Fig. 2.
figure 2

Dynamic roughness parameter, determined by (9), (24), and (25), depending on wind speed U.

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The parameters of roughness for the momentum, heat, and moisture at medium winds averaged 0.0006, 0.000073, and 0.000069 m, respectively. Figure 3a shows the dependence of roughness parameters for the temperature and momentum on wind speed. Note that at weak wind, the roughness parameters for the temperature and humidity are not equal (Fig. 3b). The ratio of the roughness scales for temperature and moisture are far greater than unit for a weak wind and decrease to values near unit for wind above the threshold value ⁓3.0 m/s. The more efficient transfer of sensible heat than latent heat at weak winds \({{z}_{{0H}}} > {{z}_{{oT}}}\) is due to the fact that thermal flow plays a considerable role in the formation of the buoyancy flux.

Fig. 3.
figure 3

(a) Dependence of roughness parameters for temperature and humidity, determined by formulas (10) and (11), on wind speed, (b) the ratio of the roughness parameter for temperature to the roughness parameter for humidity as a function of wind speed U.

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The parameterizations obtained by the authors of this study from measurements are as follows:

$${{z}_{{0u}}} = \max \left( {0.03\frac{{u_{*}^{2}}}{g},\frac{{0.135{\kern 1pt} \nu }}{{{{u}_{*}}}}} \right),$$
(34)
$${{z}_{{ot}}} = {{z}_{0}}\left\{ { - 0.56(4\sqrt {\operatorname{Re} } - 3.4} \right\},$$
(35)
$${{z}_{{oh}}} = {{z}_{0}}\left\{ { - 0.6(4\sqrt[{}]{{\operatorname{Re} }} - 3.6} \right\}.$$
(36)

The coefficients in formulas (34)–(36) are close to the coefficients obtained before [62], thus demonstrating the manifestation of a single mechanism of transport processes in the viscous sublayer of closed water bodies.

CONCLUSIONS

This study of the roughness parameters of shallow water bodies uses the data of pulsation observations in different water areas. The results show that Charnock formula can be used to calculate the parameter of dynamic roughness for shallow closed water bodies. At medium wind speeds, the results of calculations by Charnock formula coincide with observation results; in this case, the Charnock parameter is three times as high as that for the open ocean. At weak winds, the roughness forms by the surface tension or small-scale capillary waves, and the Charnock relationship in the form (21) is inapplicable, because the effect of gravity winds decreases. In this case, the passage from a viscous mechanism to a wave mechanism takes place at wind speed higher than that in the ocean. At medium winds in the lake, the dynamic roughness is greater than in the open ocean. This is likely due to the fact that waves in the lake do not reach a large age and inherit the parameters of young waves under stronger wind.

The roughness parameters for the temperature and humidity at wind speed <3 m/s are not equal. The more efficient transport of sensible heat than latent heat at weak winds is due to the fact that thermal flow plays an important part in the formation of buoyancy flux. The numerical coefficients in the dependencies of the ratio of dynamic roughness to the roughness parameter for the temperature (humidity) on the Reynolds number are close to the values obtained before for other closed water bodies, thus demonstrating that the formation mechanism of the transport processes in the viscous sublayer is the same. The obtained parameterizations can be used in climate and lake models to calculate turbulent fluxes over continental water bodies.