Modelling of the Evolving Stable Boundary Layer

Abstract

A single-column model of the evolving stable boundary layer (SBL) is tested for self-similar properties of the flow and effects of ambient forcing. The turbulence closure of the model is diagnostic, based on the K-theory approach, with a semi-empirical form of the mixing length, and empirical stability functions of the Richardson number. The model results, expressed in terms of local similarity scales, are universal functions, satisfied in the entire SBL. Based on similarity expression, a realizability condition is derived for the minimum allowable turbulent heat flux in the SBL. Numerical experiments show that the development of “horse-shoe” shaped, fixed-elevation hodographs in the interior of the SBL around sunrise is controlled by effects imposed by surface thermal forcing.

1 Introduction

Significant interest in the structure and properties of the stable boundary layer (SBL), observed during recent years, resulted in a number of field programs (e.g., Cuxart et al. 2000; Kustas et al. 2004; Schwarz et al. 2004; Oncley et al. 2007; Thomas et al. 2008), major field experiments, such as SHEBA (e.g., Andreas et al. 1999; Persson et al. 2002; Grachev et al. 2005), and CASES-99 (e.g. Blumen et al. 2001; Poulos et al. 2002), as well as numerical model intercomparisons (e.g., Beare et al. 2006; Cuxart et al. 2006) and investigations (e.g., Sun et al. 2006; Sanz Rodrigo and Anderson 2013). The field observations showed that an ample understanding of the stably stratified boundary layer depends on high-resolution vertical measurements. Instantaneous turbulent flows in stable conditions contain thin layers with large, positive and negative vertical gradients of the potential temperature and wind velocity (e.g., Sorbjan and Balsley 2008). The SBL often does not reach equilibrium (e.g., Sun et al. 2004), is sensitive to terrain inclination, which can lead to the development of a variety of density currents (e.g., Derbyshire and Wood 1994; Monti et al. 2002) and shallow drainage flows (e.g., Mahrt et al. 2001; Shapiro and Fedorovich 2009). It is often affected by baroclinicity (Kim and Mahrt 1992), subsidence (Mirocha and Kosovich 2010), surface heterogeneity (e.g., Nappo 1991), and radiative effects (e.g., Duynkerke 1999; Sun et al. 2003; Ha and Mahrt 2003).

Despite the described complications, numerical studies, involving both large-eddy simulation (LES) and single-column models (SCM), produce reasonable results in weakly stable conditions (Kosovic and Curry 2000; Beare et al. 2006; Stoll and Porté-Agel 2008). Thus, the primary purpose of this study is to further investigate properties of the SCM approach as a key to the understanding of nocturnal turbulence. Specifically, we focus on the consistency of turbulence parametrization, self-similar properties of the flow, and the effects of ambient forcing.

The paper has the following structure: model equations and assumptions are discussed in Sect. 2 and model results are compared with large-eddy simulations in Sect. 3. In the same section, the universal functions of the Richardson number are derived based on the turbulence closure equations. The effects of the surface forcing are discussed in Sect. 4, and final remarks are provided in Sect. 5.

2 Model

2.1 Governing Equations

The presented model is an improved version of the single-column model (SCM) developed by Sorbjan (2012b). It employs the ensemble-averaged, horizontally-homogeneous equations for the wind components and temperature in the atmospheric boundary layer of the form,

$$\begin{aligned} \frac{\partial U}{\partial t}&= f(V-V_\mathrm{g} )-\frac{\partial \tau _x }{\partial z}+s_\mathrm{u} ,\end{aligned}$$

(1a)

$$\begin{aligned} \frac{\partial V}{\partial t}&= -f(U-U_\mathrm{g} )-\frac{\partial \tau _y }{\partial z}+s_\mathrm{v} ,\end{aligned}$$

(1b)

$$\begin{aligned} \frac{\partial \varTheta }{\partial t}&= -\frac{\partial H}{\partial z}+s_\theta , \end{aligned}$$

(1c)

where $U$, $V$ are the horizontal components of the wind vector, ${U}_\mathrm{g}$, ${V}_\mathrm{g}$ are the components of the geostrophic wind, $f$ is the Coriolis parameter, $\varTheta$ is the potential virtual temperature, $\tau _{x}$, $\tau _{y}$, and $H$ are the turbulent fluxes for momentum and temperature, $z$ is the height, and $t$ is time. The first two equations of the system above are the momentum budgets, and the third equation follows from the budget of the virtual potential temperature. The terms ${s}_\mathrm{u}$, ${s}_\mathrm{v}$, and ${s}_{\theta }$ express the effects of external forcing (which can include the effects of baroclinicity, radiation, subsidence, etc.).

The turbulent fluxes in Eq. 1 are evaluated based on K-theory (Prandtl 1932),

$$\begin{aligned} \tau _x&= -K_\mathrm{m} \frac{\mathrm{d}U}{\mathrm{d}z} ,\end{aligned}$$

(2a)

$$\begin{aligned} \tau _y&= -K_\mathrm{m} \frac{\mathrm{d}V}{\mathrm{d}z} ,\end{aligned}$$

(2b)

$$\begin{aligned} H&= -K_\mathrm{h} \frac{\mathrm{d}\varTheta }{\mathrm{d}z}, \end{aligned}$$

(2c)

where ${K}_\mathrm{m}$ and ${K}_\mathrm{h}$ are the eddy diffusivities, assumed to be of the form

$$\begin{aligned} K_\mathrm{m}&= l^{2} S f_\mathrm{m},\end{aligned}$$

(3a)

$$\begin{aligned} K_\mathrm{m}&= l^{2} S f_\mathrm{h}, \end{aligned}$$

(3b)

where $l$ is the mixing length in neutral conditions, $S$ is the wind shear, and ${f}_\mathrm{m}$ and ${f}_\mathrm{h}$ are empirical functions of the Richardson number, defined as follows,

$$\begin{aligned} f_\mathrm{m}&= \frac{1}{(1+300Ri^{2})^{3/2}},\end{aligned}$$

(4a)

$$\begin{aligned} f_\mathrm{h}&= \frac{1}{0.9 (1+250Ri^{2})^{3/2}}. \end{aligned}$$

(4b)

Above ${Ri} = {N}^{2}/{S}^{2}$ is the Richardson number, ${N} = (\beta \varGamma )^{1/2}$ is the Brunt–Väisälä frequency, $\beta = g/T_\mathrm{o}$ is the buoyancy parameter, and $\varGamma = \mathrm{d}\varTheta /{\mathrm{d}z}$ is the actual gradient of the potential temperature. The form of the above stability functions was obtained based on SHEBA observations in the surface layer (Sorbjan 2010; Sorbjan and Grachev 2010) for $Ri < 0.7$.

Equation 3a follows from an expressions for the eddy viscosity ${K}_\mathrm{m} = l^{2}{S}$ of Prandtl (1932), derived under the assumption that thermal stratification is neutral. The mixing length characterizes dominant eddies, and is formally defined as $l = \tau ^{1/2}/{S}$, where $\tau = {(\tau _{x}^{2} + \tau _{y}^{{2}})}^{1/2}$. In close proximity of the underlying surface, observations show that $S = u_{*}/(\kappa z)$, where $u_{*} = \tau ^{1/2}$ is the friction velocity, and $\kappa = 0.4$ is the von Karman constant. This implies that the mixing length in the neutral surface layer is a linear function of height $l = \kappa z$.

Farther from the surface, the growth of the mixing length with height is expected to be gentler (Sorbjan 2012a). This fact can be taken into consideration by adopting the inverse linear approximation of Delage (1974), between the limits of the mixing length near the underlying surface $(= \kappa z)$ and in the neutral boundary layer $(= \lambda _\mathrm{o}): 1/l = 1/(\kappa z) + 1/(\lambda _\mathrm{o})$, which is equivalent to $l = z /(1+ \kappa z/\lambda _\mathrm{o})$. Blackadar (1962) suggested that $\lambda _\mathrm{o} = 0.009 u_{*}/f$, where $f$ is the Coriolis parameter. Following Huang et al. (2013), we employ an additional modification of the mixing length by proposing the inverse linear approximation of the form: $1/l = 1/(\kappa z) + 1/\lambda _\mathrm{m}$, where $1/ \lambda _\mathrm{m} = 1/\lambda _\mathrm{o} + 1/\lambda _\mathrm{B}$, and $\lambda _\mathrm{B}$ is a stability parameter, which takes into consideration that the boundary layer is topped by the stably-stratified free atmosphere. This yields,

$$\begin{aligned} l = \frac{\kappa z}{1 + \frac{\kappa z}{\lambda _\mathrm{o}} (1 + \lambda _\mathrm{o}/\lambda _\mathrm{B} )}. \end{aligned}$$

(5)

Huang et al. (2013) concluded, based on a series of LES, that

$$\begin{aligned} \lambda _\mathrm{B} = \lambda _\mathrm{s}/ Ri, \end{aligned}$$

(6)

where $\lambda _\mathrm{s}= 0.27\,\hbox {m}$, and we will assume that $\lambda _\mathrm{s}= 1 \,\hbox {m}$. Note that when, $\lambda _\mathrm{o}/\lambda _\mathrm{B} \rightarrow 0$ then $l \rightarrow \kappa z/(1 + \kappa z /\lambda _\mathrm{o})$.

The stability correction in Eq. 5 causes the mixing length l to be reduced in the upper portion of the SBL. For example, for $u_{*} \approx 0.3 \,\hbox {m}\, \hbox {s}^{-1}, f = 10^{-4} \,\hbox {s}^{-1}$, and $Ri = 0.2$, we obtain $\lambda _\mathrm{o} \approx 33 \,\hbox {m}, \lambda _\mathrm{B}=\lambda _\mathrm{s}/Ri = 5 \,\hbox {m}$, and $\lambda _\mathrm{o}/(1+\lambda _\mathrm{o}/\lambda _\mathrm{B})\approx 4.3 \,\hbox {m}$. In addition, we will also assume that for $Ri > 0.7$ turbulence vanishes, and $l = 0$.

It should be noted that the turbulence parametrization Eqs. 2–6 is diagnostic, i.e. it is not directly dependent on time, and also that turbulence is characterized by the mean-state parameters in the entire range of eddies. Its intensity results from the local competition between the vertical wind shear and the stabilizing effects of the background buoyancy. The wind shear $S$ supplies the kinetic energy for turbulent motion, while buoyancy, expressed by the Brunt–Väisälä frequency $N$, acts to restrain it. If $S^{2} > N^{2}$ (i.e. $Ri < 1$), shear production dominates and turbulence is amplified. Otherwise, buoyancy prevails and turbulence is suppressed. The system has one internal governing stability parameter, the Richardson number Ri, which dynamically adjusts to the boundary conditions, and to the external forcing. The profiles $S^{2}$ and $N^{2}$ are smooth, without fluctuations observed in the atmosphere (e.g. Sorbjan and Balsley 2008). Thus, the effect of turbulent mixing is assumed to be represented by the mixing length.

The semi-empirical formulation of the mixing length, Eqs. 5 and 6, has a shortcoming related to the fact that $l$ is not local. The mixing length locally depends on the Richardson number Ri, but it is also a prescribed function of the height z. Such a deficiency can be improved by considering more advanced closure schemes. Sorbjan and Czerwinska (2013) found that Eqs. 5 and 6 do not work well in more complex cases, and that improvement can be achieved by assuming that $l \approx \sigma _\mathrm{w}/N$, or $l \approx \beta \sigma _{\theta }/N^{2}$, where $\sigma _\mathrm{w}$ and $\sigma _{\theta }$ are the standard deviations of the vertical velocity and temperature.

2.2 Initial and Boundary Conditions

The system (1)–(6) is augmented by the no-slip boundary conditions at the surface, and the free-slip conditions at the top of the vertical domain,

$$\begin{aligned} \hbox {at}\, z&= z_\mathrm{o}:\; U = 0, V = 0, \varTheta = \varTheta _\mathrm{o}- C_\mathrm{R} t,\end{aligned}$$

(7a)

$$\begin{aligned} \hbox {at}\, z&= D: \; \mathrm{d}U/\mathrm{d}z = 0, \mathrm{d}V/\mathrm{d}z = 0, \mathrm{d}\varTheta /\mathrm{d}z = \gamma , \end{aligned}$$

(7b)

where $z_\mathrm{o}$ is the aerodynamic roughness length, $D$ is the vertical extent of the considered domain, $\varTheta _\mathrm{o}$ is the initial surface temperature, $\gamma$ is the value of the potential temperature gradient above the boundary layer, $C_\mathrm{R}$ is the cooling rate, and $t$ is time.

The adopted initial condition formulation follows the GEWEX Atmospheric Boundary layer Study (GABLS) intercomparison set-up (Cuxart et al. 2006). It defines a neutrally stratified layer with a potential temperature $\varTheta _\mathrm{o}= 265 \,\hbox {K}$, extending up to 100 m; above the 100 m layer, the potential temperature increases according to the gradient $\gamma = 0.01 \,\hbox {K m}^{-1}$. The vertical domain of the model is 2,910 m. The x-axis of the coordinate system is aligned with the geostrophic wind vector, such that $U_\mathrm{g}= G, V_\mathrm{g}= 0$, with the geostrophic wind $G$ set to $8 \,\hbox {m s}^{-1}$. The Coriolis parameter is $f = 1.39 \times 10^{-4} \,\hbox {s}^{-1}$, which corresponds to latitude $73^{\circ }\hbox {N}$, and we take $z_\mathrm{o} = 0.1 \,\hbox {m}$. The initial flow is assumed geostrophic. In the basic set-up of the model, a prescribed surface cooling rate of $C_\mathrm{R}= 0.25\, \hbox {K h}^{-1}$ is applied for 9 or 24 h. In the other two runs, the cooling rate was applied for the first 8 h, followed by 1- or 2-h period of warming with $C_\mathrm{R}= - 0.25 \,\hbox {K h}^{-1}$. The external forcing $s_\mathrm{u}, s_\mathrm{v}, s_{\theta }$ is not considered herein.

It could be noted that, when $C_\mathrm{R}= 0$ and $\gamma > 0$, the resulting boundary layer is stable. Turbulence generated by shear causes the upper part of the SBL to be cooled, and the lower part to be slightly warmed. For the pure neutral case to exist, it is required that $C_\mathrm{R}= 0$ and $\gamma = 0$. When the cooling rate $C_\mathrm{R}$ increases, the depth of the SBL becomes smaller.

2.3 Numerical Solution

The model equations were solved numerically by using the method described in the Appendix. In order to obtain sufficient accuracy and to resolve the logarithmic sublayer, the grid spacing was very fine and non-uniform (logarithmic), with grid points located at $z_{j} = 10^{\left[ \zeta _\mathrm{o} + (j - 1)\delta z \right] }$ where $\zeta _\mathrm{o} = \log _{10}(z_\mathrm{o}), \delta z$ is the grid increment, assumed to be equal to 0.018. The time increment was also small, equal to 0.05 s. The number of grid points was typically 250, with the grid increment along the vertical coordinate changed from a fraction of $z_\mathrm{o}$ near the surface, to about 120 m near the upper boundary. The unknown values $U$, $V$, and $\varTheta$ were defined at levels $z_{j}$, and the eddy diffusivities $K_\mathrm{m}$, $K_\mathrm{h}$, temperature gradient $\varGamma$, shear $S$, and fluxes at midpoints $z_{j\pm 1/2}$.

The fine resolution ensures that the Monin–Obukhov similarity is exactly satisfied in the lower portion of the model domain. Consequently, the model does not require evaluation of the surface fluxes by inverting the Monin–Obukhov similarity expressions, as done in coarse-resolution SCMs. The surface fluxes can be directly obtained from Eqs. 2–5, by using gradients calculated for the first two grid points. A linear grid can also be used in the model for practical purposes, which requires applying the Monin–Obukhov similarity theory near the underlying surface in order to derive fluxes based on gradients. In this case, the number of grid points can be significantly smaller.

3 Results

3.1 Vertical Profiles

The vertical profiles of the potential temperature, wind speed, and turbulent fluxes, obtained from the 9-h basic run of our SCM, are shown in Fig. 1a–c. The results from high-resolution LES, performed at the University of Hannover, are also presented (Raash and Etling 1991; Raash and Schröter 2001). The LES and the SCM runs were performed for analogous initial conditions and forcing. The SCM results are presented as solid lines in the figure, and agree well with the LES results, which are represented by dashed red lines.

Fig. 1

Model results obtained in the 9-h basic run: a the potential temperature, b the wind-velocity modulus, c the negative heat flux $-H$ and the modulus of the momentum flux $\tau$. Profiles obtained in the University of Hannover LES simulations (Raash and Etling 1991; Raash and Schröter 2001) are indicated by red dashed lines

The potential temperature for the 9-h basic run is depicted in Fig. 1a together with the initial condition. Results of both models show the surface temperature of about 263 K, and similar profile curvatures in the boundary layer. The local wind maximum (low level jet) in Fig. 1b is about $9.5 \,\hbox {m s}^{-1}$, and takes place at $z$ = 140 m. Both turbulent fluxes, $-H$ and $\tau$, decrease with height (in Fig. 1c), and reach relatively small values at levels above 150 m. the results of the current version of the model (Fig. 1) are not significantly different from the previous version (Fig. 2 in Sorbjan 2012b). Nonetheless, the new formulation shows that the SCM very closely reproduces now the LES curves in Fig. 1.

The main difference between the SCM presented here and other SCMs, research and operational (e.g., Duynkerke 1991; Kim and Mahrt 1992; Galmarini et al. 1998; Beare and Mac Vean 2004; Edwards et al. 2006), is in the form of the stability functions $f_\mathrm{m}$ and $f_\mathrm{h}$, and the adjustable values of $\lambda _\mathrm{m}$ (see Sect. 2.1). The values of $\lambda _\mathrm{m}$, for the models included in the GABLS intercomparison, varied between 40 and 200 m. Generally, operational models utilize a pragmatic form of $f_\mathrm{m}$ and $f_\mathrm{h}$, which slowly decay with $Ri$, and cause higher mixing efficiencies than would be expected based on local observations. As a consequence, such models avoid an excessive cooling near the surface over land points, which improves operational verification scores, but is not physically realistic.

Figure 2a depicts profiles of the Brunt–Väisälä frequency $N$ (red line), shear $S$ (blue line), and the Richardson number $Ri$ (black line). Referring to the thermal stability classification of Sorbjan (2010), based on the values of the Richardson number, one can verify in the figure that:

• (i) the “near-neutral” regime $(0 < Ri < 0.02)$ is located within the first 3 m above the ground,

  Fig. 2

  

  Profiles of: a the Brunt–Väisälä frequency $N$, shear $S$, and the Richardson number $Ri$, b the mixing length $l$ and the Ozmidov length $l_\mathrm{o}=\varepsilon ^{1/2}/N^{3/2}$, obtained in the 9-h basic run. The Ozmidov length $l_\mathrm{o}$ was estimated based on Eq. 13a

• (ii) the “stable” layer $(0.02 < Ri < 0.12)$, extends from 3 m to about 108 m,

• (iii) the “very stable” layer $(0.12 < Ri < 0.7)$, stretches from 108 to 180 m,

• (iv) the “extremely stable” layer $(Ri > 0.7)$ is located above the level of 180 m.

Figures 1 and 2 indicate that, at the level $z = 180 \,\hbox {m}$, which can be considered as the height of the boundary layer, there is no cooling effect with respect to the assumed initial temperature, the wind velocity is geostrophic, and the turbulent fluxes are practically zero. The Brunt–Väisälä frequency has a local maximum, and the Richardson number $Ri$ sharply increases with height. Figure 2b shows profiles of the mixing length $l$ and the Ozmidov length $l_\mathrm{o}=\varepsilon ^{1/2}/N^{3/2}$, where the dissipation rate $\varepsilon$ was evaluated based on Eq. 12a below. The Ozmidov length is the outer scale of isotropic turbulence in stable stratified flows. Within the first 10 m above the surface, the mixing length varies linearly, and above the surface layer the increase of the mixing length is significantly reduced. At scales smaller than $l_\mathrm{o}$, buoyancy has only a minor damping effect on isotropic turbulence, but dominates at larger ones. Consequently, based on Fig. 2b, we expect that below the level of 120 m, where $l_\mathrm{o} < l$, turbulence is expected to be isotropic, and anisotropic above this level.

3.2 Universal Formulation Within the Model

Sorbjan (2010) introduced the following local scales for velocity, temperature, length:

$$\begin{aligned} U_{N}&= l\, N, \end{aligned}$$

(8a)

$$\begin{aligned} T_{N}&= l\, \varGamma =l N^{2}{/}\beta , \end{aligned}$$

(8b)

$$\begin{aligned} L_{N}&= l. \end{aligned}$$

(8c)

The above scales involve the Brunt–Väisälä frequency $N$, and therefore will be referred to as the “gradient-based $N$-scales”. In terms of the above scales, the following expressions can be directly obtained based on (2)–(4),

$$\begin{aligned} \frac{\tau }{U_N}&= \frac{f_\mathrm{m}}{Ri}=\frac{1}{Ri(1+300Ri^{2})^{3/2}},\end{aligned}$$

(9a)

$$\begin{aligned} -\frac{H}{U_N T_N }&= \frac{f_\mathrm{h}}{Ri^{1/2}}=\frac{1}{0.9Ri^{1/2}(1+250Ri^{2})^{3/2}},\end{aligned}$$

(9b)

$$\begin{aligned} Pr&= \frac{K_\mathrm{m}}{K_\mathrm{h}} =\frac{f_\mathrm{m}}{f_\mathrm{h}}= \frac{Ri}{Rf} = 0.9\frac{{(1 + 250Ri^{2})}^{3/2}}{{(1 + 300Ri^{2})}^{3/2}} \end{aligned}$$

(9c)

where $Rf = - \beta H/(\tau S)$ is the flux Richardson number, and $Pr$ is the Prandtl number. Note that Eq. 9c indicates that $Pr < 1$ (Sorbjan and Grachev 2010).

On the other hand, one can also obtain:

$$\begin{aligned} \varPsi _\mathrm{m}&\equiv \frac{l_ S}{U_*}=\frac{1}{f_\mathrm{m} ^{1/2}}=(1+300Ri^{2})^{3/4},\end{aligned}$$

(10a)

$$\begin{aligned} \varPsi _\mathrm{h}&\equiv \frac{l_ \varGamma }{\varTheta _*}=\frac{f_\mathrm{m} ^{1/2}}{f_\mathrm{h} }=0.9\frac{(1+250Ri^{2})^{3/2}}{(1+300Ri^{2})^{3/4}},\end{aligned}$$

(10b)

$$\begin{aligned} \frac{l}{\varLambda _*}&= \frac{\kappa }{Ri}\frac{f_\mathrm{m} ^{3/2}}{f_\mathrm{h} }=\frac{\kappa }{0.9}\frac{Ri(1+300Ri^{2})^{9/4}}{(1+250Ri^{2})^{3/2}}, \end{aligned}$$

(10c)

where the flux-based local scales are of the form:

$$\begin{aligned} U_{*}&= \tau ^{1/2}, \end{aligned}$$

(11a)

$$\begin{aligned} \varTheta _{*}&= H/ U_{*}, \end{aligned}$$

(11b)

$$\begin{aligned} \varLambda _{*}&= - \tau ^{3/2}/ (\kappa \beta H). \end{aligned}$$

(11c)

Contrary to the Monin–Obukhov scales $u_{*},T_{*}, L_{*}$, the above scales in Eq. 11 are height-dependent.

Because the turbulence parameterization is diagnostic, the expressions in Eqs. 9–10 are exactly satisfied within the model, for each moment of time, for each level in the SBL, for all any applied external forcing. Equations 10a–c extend the surface-layer (Monin–Obukhov) similarity formulation for the entire SBL. We note that the dimensionless expressions (9)–(10) are not independent of each other, and all are formulated in terms of two similarity functions $f_\mathrm{m}$, $f_\mathrm{h}$ of the Richardson number $Ri$, defined by (4).

In the quasi-steady state, characteristics of turbulence do not significantly vary with time (see Fig. 6a). Consequently, budgets for the turbulent energy and for the temperature variance, can be approximately expressed in the following form,

$$\begin{aligned}&K_\mathrm{m} S^{2} (1 - Rf) = \varepsilon ,\end{aligned}$$

(12a)

$$\begin{aligned}&\varepsilon _{\theta } = K_\mathrm{h}\varGamma ^{2}, \end{aligned}$$

(12b)

where $\varepsilon$ is the dissipation rate, and $\upvarepsilon _{\uptheta }$ is the dissipation for one-half of the temperature variance. Using Eqs. 3, 4, and 9, yields

$$\begin{aligned}&\displaystyle \frac{\varepsilon }{U_N ^{2}/t_{s}}=\frac{f_\mathrm{m} }{Ri^{3}}(1-Rf)=\frac{1}{Ri^{3}(1+300Ri^{2})^{3/2}} \left[ {1-\frac{Ri}{0.9}\frac{(1+300Ri^{2})^{3/2}}{(1+250Ri^{2})^{3/2}}} \right] ,\qquad \quad \end{aligned}$$

(13a)

$$\begin{aligned}&\displaystyle \frac{\varepsilon _\theta }{{T_N}^{2}/t_{N}} = \frac{f_\mathrm{h}}{Ri^{1/2}}= \frac{1}{0.9Ri^{1/2} (1 + 250Ri^{2})^{3/2}}. \end{aligned}$$

(13b)

Since the dissipation rate $\varepsilon$ is a positive-definite quantity, Eq. 13a allows us to conclude that the steady state, which results from a balance of shear production and buoyant-dissipative destruction, takes place only for $Rf <1$. Equation 9c indicates that $Rf = 1$ when $Ri = Ri_\mathrm{s}= 0.7$ (Sorbjan 2010). Thus, the steady-state turbulence would not be present at values of $Ri$ exceeding $Ri_\mathrm{s} = 0.7$, which is larger than the critical value ${ Ri}_\mathrm{cr} = 0.25$, implied by linear stability theory. In other words, at $Ri > Ri_\mathrm{s}$, turbulence is expected to be non-stationary, i.e., decaying or sporadic. The inequality $Ri < Ri_\mathrm{cr} = 0.25$ is a sufficient condition for the presence of steady-state turbulence, i.e., if satisfied, it guarantees that steady-state turbulence exists. The inequality $Ri < Ri_\mathrm{s} = 0.7$ is a necessary condition for the presence of steady-state turbulence, i.e., it must be satisfied for steady-state turbulence to occur. Thus, the stability functions (4) for $f_\mathrm{m}$ and $f_\mathrm{h}$ are assumed valid for $Ri < 0.7$. Nonetheless, Grachev et al. (2013) argued that turbulence in the inertial range decays at $Ri > 0.25$.

The dimensionless similarity functions obtained at the end of the 9-h basic run are examined in Fig. 3 for the momentum flux $G_\mathrm{t}=\tau /U_{s}$, heat flux $G_\mathrm{h}= - H/(U_{s}T_{s})$, wind shear $\varPsi _\mathrm{m} = l \,S /U_{*}$, temperature gradient $\varPsi _\mathrm{h}= l \,\varGamma /\varTheta _{*}$, and for the ratio $l/(\kappa \varLambda _{*})$ of the mixing length and the local Obukhov length. The lines in the figure indicate the numerical results, while the dots mark the values obtained from the analytical expression (9)–(10). As expected, the agreement is exact, i.e., the expressions (9)–(10) are exactly satisfied within the model for each moment of time, for each level in the SBL, and for all any external forcing applied in the model.

Fig. 3

Dimensionless functions obtained in the 9-h basic run for: a momentum flux $G_\mathrm{t}=\tau /U_{s}$ and heat flux $G_\mathrm{h }= - H/(U_{s}T_{s})$, b wind shear $\varPsi _\mathrm{m} = l\, S/U_{*}$ and temperature gradient $\varPsi _\mathrm{h}= l\,\varGamma /\varTheta _{*}$, c the ratio of the mixed length and the local Obukhov length, $l\, /\varLambda _{*}$. Lines indicate the model results, while the red dots mark the values obtained from the analytical expression (9)–(10). The agreement of the numerical results and analytical values is exact

3.3 The Temperature-Flux Realizability Condition

Let us consider another gradient-based scaling system of the following form,

$$\begin{aligned} U_{S}&= l\, S, \end{aligned}$$

(14a)

$$\begin{aligned} T_{S}&= l\, S^{2}/\beta ,\end{aligned}$$

(14b)

$$\begin{aligned} L_{S}&= l. \end{aligned}$$

(14c)

The above scales involve the shear S, and therefore will be referred to as the “gradient- based S-scales”. It can be readily verified that

$$\begin{aligned} U_{S}&= U_{N} Ri^{-1/2},\end{aligned}$$

(15a)

$$\begin{aligned} T_{S}&= T_{N} Ri^{-1}. \end{aligned}$$

(15b)

Consequently, the expressions (9b) can be rewritten as

$$\begin{aligned} -\frac{H}{U_S T_S }=Ri f_\mathrm{h} =\frac{Ri}{0.9(1+250Ri^{2})^{3/2}}. \end{aligned}$$

(16)

Figure 4 shows the dimensionless temperature flux $- H/(U_{S} T_{S})$ obtained in the model as a function of $Ri$, and also as a function of height $z$. Figure 4a indicates that the dimensionless flux $- H/(U_{S} T_{S})$ vanishes in neutral conditions at $Ri$ = 0, and also in the very stable case, where $Ri$ is large. It has a minimum where the first derivative with respect to $Ri$ is zero. It can be readily verified based on Eq. 16 that the minimum, equal to $[0.9 \times 500^{1/2}(1.5)^{3/2}]^{-1}= - 0.027$, is reached at $Ri_{\min } = (500)^{-1/2}= 0.0447$. Taking this result into consideration, we obtain

$$\begin{aligned} H_{\min } (z)=-0.027 U_S T_S =-0.027 \frac{l^{2}_\mathrm{m}(z)S^{3}(z)}{\beta }, \end{aligned}$$

(17)

which constitutes the realizability condition for the heat flux at a specific level $z$. The inequality $H(z) \ge H_{\min }(z)$ must be satisfied at any height $z$ in the SBL. An expressions analogous to Eq. 17 was derived by Van de Wiel et al. (2011) based on Monin–Obukhov similarity theory in the surface layer; they found that the minimum of the temperature flux occurs at a larger value of $Ri_{\min } = 0.07$.

It can be noted that for $Ri_{\min } = 0.0447$, one obtains from Eq. 10c that $(l/\varLambda _{*})_{\min } = 0.0311$, which implies that, in the surface layer where $l \approx \kappa z$, the parameter $(z/L_{*})_{\min } = 0.0781$. The bin-averaged temperature flux, measured at five levels during the SHEBA experiment, show a similar minimum at $z/ L_{*}$ in the range $10^{-2} - 10^{-1}$ (A.A. Grachev, private communication, 2012).

Fig. 4

Profiles of the temperature flux obtained in the basic run of the model: a the dimensionless temperature flux $-H/(U_{S}T_{S})$ as a function of the Richardson number $Ri$ (the dots indicate the values plotted based on Eq. 16—the agreement of the numerical results and analytical values is exact), b the dimensionless temperature flux $-H/(U_{S}T_{S})$ as a function of height $z$, c the actual temperature flux $H$ and the minimum flux $H_{\min }$ (see Eq. 17) as a function of height. Note that $H$ $\ge H_{\min }$, and that $H = H_{\min }$ at $z = 11 \,\hbox {m}$, and at $z =180 \,\hbox {m}$

In Fig. 4a, the values of the dimensionless heat flux $- H/(U_{S}T_{S})$, obtained from the model, are plotted versus $Ri$; the dots indicate the values plotted based on Eq. 16. The agreement of the numerical results (curves) and the analytical values described by Eq. 16 is exact. In Fig. 4b, the values of the dimensionless heat flux $-H/(U_{S}T_{S})$ are plotted versus height; the minimum value of the dimensionless flux occurs at about $z = 11 \,\hbox {m}$. Figure 1c implies that, at $z = 11 \,\hbox {m}$, the heat flux $H = - 0.0075 \,\hbox {K m s}^{-1}$; using Eq. 17, with $S = 0.1053\,\hbox {s}^{-1}$ (see Fig. 2a), $l = 2.97 \,\hbox {m}$ (see Fig. 2b), and $\beta = 0.0374\, \hbox {m s}^{-2}$ $\hbox {K}^{-1}$, we obtain the same value.

In Fig. 4c, the actual heat flux $H$ and the minimum permissible heat flux $H_{\min }$ are plotted as functions of height. Values of both functions, $H$ and $H_{\min }$, are equal at $z$ = 11 m; below and above this level, the actual heat flux is larger that the permissible one. At the top of the SBL both values are equal to zero.

4 Temporal Evolution

Next, we discuss the temporal evolution of the nocturnal boundary layer, imposed by surface forcing, and also by the inertial effects. The inertial mechanism was first described by Blackadar (1957), as a phenomenon in the upper (nearly frictionless) part of the boundary layer. It involves undamped oscillations of wind vector around the geostrophic wind vector (Buajitti and Blackadar 1957). The phenomenon has a period of $T_{i} = 2\pi /f$, and is commonly observed in the atmosphere (e.g., Poulos et al. 2002). Inertial oscillations are considered as an important mechanism behind the occurrence of nocturnal low-level jets (e.g., Banta 2008).

Let us begin with a discussion of the nocturnal evolution of the boundary layer, provided by the model results. Figure 5 shows profiles of the potential temperature and the temperature flux, obtained in three model runs. In the first run, the underlying surface was cooled for 9 h at a rate $C_\mathrm{R} = 0.25 \,\hbox {K h}^{-1}$. In the second run, the 8-h cooling was followed by 1-h warming with the rate $C_\mathrm{R}= - 0.25 \,\hbox {K h}^{-1}$. In the third run, the 8-h cooling was followed by 2-h warming with the same rate of $C_\mathrm{R}= - 0.25 \,\hbox {K h}^{-1}$. The runs with warming intervals are intended to imitate the sunrise transition. The figure shows that when surface warming takes place, the temperature profiles are reshaped. They become more uniform with height, due to warming near the surface, while a strong temperature inversion layer forms at the top of the boundary layer. The heat flux, which is nearly linear in the cooling case, becomes strongly curved due to surface heating.

Fig. 5

Profiles of a the potential temperature, b the temperature flux, obtained after 8-h cooling with $C_\mathrm{R} = 0.25\, \hbox {K h}^{-1}$, followed by: (i) additional 1-h cooling (dotted lines), (ii) 1-h warming with $C_\mathrm{R} = - 0.25\, \hbox {K h}^{-1}$ (blue lines), (iii) 2-h warming with $C_\mathrm{R} = - 0.25\, \hbox {K h}^{-1}$ (red lines)

The corresponding time evolution of the drag coefficient $u_{*}/G$, and the cross-isobar angle $\alpha$, is shown in Fig. 6. The figure indicates that frictional drag rapidly decreases with time (Fig. 6a), and decreased friction allows the Coriolis force to change the wind direction (Fig. 6b). After a few hours of cooling, a quasi equilibrium is achieved, with both parameters, $u_{*}/G$ and $\alpha$ nearly constant with time. When the quasi-steady state is interrupted by the “sunrise” warming, the drag coefficient increases rapidly, and the cross-isobar angle decreases.

Fig. 6

The time evolution of: a the drag coefficient $u_{*}/G$, and b the cross-isobar angle $\alpha$, obtained after 8-h cooling with $C_\mathrm{R} = 0.25\, \hbox {K h}^{-1}$, followed by: (i) additional 2-h of cooling (black lines), (ii) 2-h warming with $C_\mathrm{R} = - 0.25\, \hbox {K h}^{-1}$ (red lines)

In Fig. 7, two “fixed-time” hodographs are presented as black lines. The first hodograph (Fig. 7a) was obtained after 9-h surface cooling of the boundary layer with the rate $C_\mathrm{R} = 0.25 \,\hbox {K h}^{-1}$. The second hodograph (Fig. 7b) is a result 8-h cooling with the rate $C_\mathrm{R}$, followed by 1-h warming with the rate $C_\mathrm{R} = - 0.25\, \hbox {K h}^{-1}$. The red lines (“fixed-elevation” hodographs) indicate how the tips of the wind vectors evolve in time at four levels: $z$ $=$ 26, 50, 101, and 153 m. It should be remembered that the initial wind is geostrophic, of magnitude $8\, \hbox {m s}^{-1}$, and directed along the $x$-axis. This is an idealization within the model, since a non-zero crosswind is usually observed during late afternoon in the atmospheric boundary layer, resulting in a cross-isobar angle of about $5^{\circ }$. The effect of the initial condition is strong during the first part of the simulation.

In Fig. 7a, b, the angle between the actual wind at a given level (marked by a red point on the fixed-elevation hodograph) and the geostrophic wind increases with time within the first few hours of the boundary-layer development, and subsequently it slightly decreases (which is consistent with Fig. 6b). As a result, the red-line hodographs turn clockwise and take on the characteristic “horseshoe” shapes. Similar hodographs were presented in Weng and Taylor (2006).

Fig. 7

Hodographs of the wind vector (black lines) obtained after: a 9-h surface cooling of the boundary layer with $C_\mathrm{R} = 0.25 \,\hbox {K h}^{-1}$, b 8-h cooling with $C_\mathrm{R} = 0.25\, \,\hbox {K h}^{-1}$, followed by 1-h warming with $C_\mathrm{R} = - 0.25\, \hbox {K h}^{-1}$. The initial wind is geostrophic with $G = 8\, \hbox {m s}^{-1}$, located along the x-axis. The red lines (the fixed-elevation hodographs) indicate the endings of the wind vectors at the levels: 26, 50, 101 m, and 153 m. The dashed lines in the right plot are obtained during an additional 1-h warming (i.e., after 8-h cooling and 2-h warming) with $C_\mathrm{R} = - 0.25\, \hbox {K h}^{-1}$

The continuous segments of the red lines in Fig. 7a were obtained at $z$ $=$ 26, 50, and 101 m, during the 9-h cooling. It should be noted that they will not change significantly even during a 24-h simulation (see the discussion below), because the quasi-steady state is reached after about 9 h. As a result, velocity components $U$ and $V$ are nearly constant with time; consequently the “horseshoe” hodographs in Fig. 7a will not become rounder. The continuous segments of the red lines in Fig. 7b were obtained at the same levels of $z$ = 26, 50, and 101 m, during the 8-h cooling and 1-h warming. The dotted segments mark the additional 1-h warming at the end of the simulation. The “warming” segments are nearly perpendicular to the continuous ones, which causes the red hodographs to appear more circular. At $z$ = 153 m, the wind speed exceeds the value of the geostrophic wind, a phenomenon referred to as the low-level jet.

To further examine the transitory issues, let us consider Fig. 8, which shows the temporal evolution of wind-vector components $U$ and $V$ at three levels: $z$ = 50 m (blue lines), 101 m (red lines), and 150 m (black lines). They were obtained in two model runs. The first run consisted of a 24-h cooling with $C_\mathrm{R} = 0.25\, \hbox {K h}^{-1}$; the results are represented by thin lines. The second run included 8-h cooling with $C_\mathrm{R} = 0.25\, \hbox {K h}^{-1}$, followed by 2-h warming with $C_\mathrm{R}= - 0.25\, \hbox {K h}^{-1}$. The results are marked by thick lines. The second run was terminated at the instant that the surface heat flux reached values near zero (as the turbulence closure applies only to cases when $Ri > 0$). Within the first 8 h the results of both runs coincide.

At $z$ $=$ 150 m, near the top of the boundary layer, inertial oscillations of both components of wind velocity develop with a period of about 13 h (note that for the value $f = 1.39 \times 10^{-4} s^{-1}$, adopted in our model, the inertial period is $T_{i} = 12.55 \,\hbox {h}$). The oscillations are damped by turbulent mixing, as the second maximum $({\approx }8.8\, \hbox {m s}^{-1})$ at $t \approx$ 22 h is noticeably lower than the first maximum $({\approx }9.5\, \hbox {m s}^{-1})$ at $t \approx$ 8 h. The oscillations at the top of the boundary layer (at $z$ = 180 m) are not damped (not shown). Within the boundary layer ($z$ = 50 and 101 m), the internal oscillations are entirely damped after about 9 h, and the velocity components reach “quasi-equilibrium”.

In the case of the 8-h cooling followed by 2-h warming, the quasi-steady state within the boundary layer is not reached. The $U$-component at $z = 50$ and 101 m begins to decrease from the moment that warming begins (such effects on the $V$-component are also noticeable but less pronounced). As a result, the three lines, representing the $U$-component at $z = 50$, 101, and 150 m during the first 10 h of the simulation are similar in Fig. 8, i.e., are nearly parallel, and have a shape of a sine function with a period of about 10 h. This value is less than the inertial period of $T_{i} =12.55\,\hbox {h}$, and consequently, these sine-like functions cannot be classified as inertial oscillations.

Fig. 8

Components of wind vectors at $z$ = 50, 101, and 150 m, obtained in two runs of the model: (i) during 24-h cooling with $C_\mathrm{R} = 0.25 \,\hbox {K h}^{-1}$ (thin lines), and (ii) 8-h cooling with $C_\mathrm{R} = 0.25\, \hbox {K h}^{-1}$, followed by 2-h warming with $C_\mathrm{R} = - 0.25 \,\hbox {K h}^{-1}$ (thick lines). The inertial period in both cases is $T_{i} =12.55 \,\hbox {h}$. Within the first 8 h, the results of both runs coincide

The above discussion implies that the development of “horse-shoe” shaped, “fixed-elevation” hodographs in the interior of the nocturnal boundary layer (in Fig. 7) are only partly caused by the damped inertial effects (as in the model of Van de Wiel et al. 2010), and are strongly affected by surface forcing due to the sunrise warming. The pure inertial oscillations occur only near the top of the SBL, as predicted by Blackadar’s (1957) theory.

5 Conclusions

Our primary purpose was to examine properties of a single-column model of the SBL, specifically the consistency of turbulence parametrization, self-similar properties, and the effects of applied forcing.

The turbulence closure of the model is diagnostic. It employs the K-theory approach, with semi-empirical formulation of the mixing length, which locally depends on the Richardson number $Ri$, but is also dependent on height. The empirical stability functions of $Ri$ are evaluated based on the SHEBA data. The model has one internal, governing stability parameter, $Ri$, which dynamically adjusts to the boundary conditions and to the external forcing. A comparison of the model results with the high-resolution, large-eddy simulations performed at the University of Hannover, reflects very good agreement for the considered GABLS case.

When expressed in terms of similarity scales, the model results are universal, i.e. they are satisfied in the entire SBL, for all time, and for the considered external forcing. Specifically, Eqs. 10a and 10b extend the surface-layer (Monin–Obukhov) similarity formulation for gradients to the entire SBL. The expressions (9)–(10), (13) for fluxes, gradients, and dissipation rates, are formulated in terms of two empirical functions $f_\mathrm{m}, f_\mathrm{h}$ of Ri.

Employing the gradient-based $S$-scales $U_{S}, T_{S}$ allows identification of the minimum of the dimensionless heat flux, with $H/(U_{S}T_{S} = - 0.027$, reached at $Ri = 0.0447$. The minimum defines the realizability limit for the heat flux in the SBL, which cannot be locally smaller than the value $H_{\min }(z) = - 0.027 l_\mathrm{m}^{2}(z) S^{3}(z)/\beta$. The numerical experiments show that the development of “horse-shoe“ shaped, fixed-elevation wind hodographs in the interior of the SBL around sunrise is controlled by surface thermal forcing, rather than by inertial effects.

Appendix: A Computational Scheme

In order to solve the system (1)–(7), a non-linear vertical mash was introduced. The unknown values $U$, $V$, $\varTheta$ were defined at levels $z_{j}$, and the eddy diffusivity at midpoints $z_{j\pm 1/2}$. This led to the following system of equations for temperature,

$$\begin{aligned} \frac{\varTheta _j^{k+1} -\varTheta _j^k }{\Delta t}=\frac{K_{j+1/2} \left( {\frac{\varTheta _{j+1}^{k+1} -\varTheta _j^{k+1} }{z_{j+1} -z_j }} \right) -K_{j-1/2} \left( {\frac{\varTheta _j^{k+1} -\varTheta _{j-1}^{k+1} }{z_j -z_{j-1} }} \right) }{\left( {z_{j+1/2} -z_{j-1/2} } \right) }+s_j^{k+1} , \end{aligned}$$

(18)

where $t = t^{k}= k \Delta t$, $j = 2, \ldots , n,\, k = 1, \ldots , m, \Delta t$ is the time increment, and $\varTheta (z, t) = \varTheta (z_{j}, t^{k}) =\varTheta _j^{k}$.

To solve the above system of linear equations, the sweeping method was used at each timestep. For this purpose, Eq. 18 was rewritten in the following form,

$$\begin{aligned} A_{j}X_{j+1} - B_{j}X_{j} + C_{j} X_{j-1} = -F_{j}, \end{aligned}$$

(19)

where for $j = 2, \ldots , n-1, X_{j}=\varTheta _{j}^{k+1}$, and $A_{j}$, $B_{j}$, $C_{j}$ and $F_{j}$ were coefficients defined as

$$\begin{aligned} A_j&= \frac{\Delta t\, K_{j+1/2}}{(z_{j+1} -z_j )(z_{j+1/2} -z_{j-1/2} )},\end{aligned}$$

(20a)

$$\begin{aligned} B_j&= 1 + A_{j} + C_{j},\end{aligned}$$

(20b)

$$\begin{aligned} C_j&= \frac{\Delta t\, K_{j-1/2} }{(z_j -z_{j-1} )(z_{j+1/2} -z_{j-1/2} )},\end{aligned}$$

(20c)

$$\begin{aligned} F_j&= \varTheta _j^k +s_j^{k+1}. \end{aligned}$$

(20d)

The boundary conditions were expressed in a general form, which is consistent with Eq. 19, viz.

$$\begin{aligned} A_{1} X_{2} + B_{1}X_{1} = F_{1}\quad \hbox {for } \,i =1,\end{aligned}$$

(21a)

$$\begin{aligned} B_{n} X_{n} + C_{n}X_{n-1} = F_{n} \quad \hbox {for }\, i = n. \end{aligned}$$

(21b)

The solution of the system of n+1 equations (18) and (21) was assumed to be in the form

$$\begin{aligned} X_{j-1} = a_{j}\,X_{j} + b_{j}, \end{aligned}$$

(22)

where for $i = n, {\ldots }, 2, a_{j}$ and $b_{j}$ are coefficients.

Note that substituting (21) into (19), yields,

$$\begin{aligned} X_{j} = \frac{A_j}{(B_j - C_j a_j)} X_{j + 1} + \frac{(F_j + C_j b_j )}{(B_{j} - C_{j} a_{j} )}. \end{aligned}$$

(23)

The above result implies that

$$\begin{aligned} a_{j+1} =\frac{A_j }{(B_j -C_j a_j )},\end{aligned}$$

(24a)

$$\begin{aligned} b_{j+1} =\frac{(F_j +C_j b_j )}{(B_j -C_j a_j )}, \end{aligned}$$

(24b)

for $i = 2, {\ldots }, n-1$. From the boundary conditions, we obtained

$$\begin{aligned} a_2 =-\frac{A_1 }{B_1 },\end{aligned}$$

(25a)

$$\begin{aligned} b_2 =\frac{F_1 }{B_1 }, \end{aligned}$$

(25b)

and

$$\begin{aligned} X_n =\frac{(F_n -C_n b_n )}{(B_n +C_n a_n )}. \end{aligned}$$

(26)

The solution can be found in two “sweeps”. First, coefficients $a_{j}$ and $b_{j}$ are found from Eqs. 25, 24, for $j$ = 2, ..., $n$. In the second sweep, $X_{n}$ can be calculated from Eq. 26, and the solution $X_{j-1}$ is obtained from Eq. 22.

The momentum equations were solved by using the matrix version of the sweeping method. For the following approximation

$$\begin{aligned} \frac{U_j^{k+1}- U_j^k }{\Delta t}&= \frac{K_{j+ 1/2} \left( {\frac{U_{j+1}^{k+1} - U_j^{k+1}}{z_{j+1} -z_j}}\right) - K_{j - 1/2} \left( {\frac{U_j^{k+1} - U_{j-1}^{k+1}}{z_j - z_{j-1}}} \right) }{\left( z_{j + 1/2} - z_{j-1/2} \right) } +f(V_j^{k + 1} - V_\mathrm{g}) + s_\mathrm{u},\end{aligned}$$

(27a)

$$\begin{aligned} \frac{V_j^{k+1} -V_j^k }{\Delta t}&= \frac{K_{j+1/2} \left( {\frac{V_{j+1}^{k+1} -V_j^{k+1} }{z_{j+1} -z_j }} \right) -K_{j-1/2} \left( {\frac{V_j^{k+1} -V_{j-1}^{k+1} }{z_j -z_{j-1} }} \right) }{\left( {z_{j+1/2} -z_{j-1/2} } \right) }-f(U_j^{k+1} -U_\mathrm{g} )+s_\mathrm{v},\nonumber \\ \end{aligned}$$

(27b)

the system (27) can be rewritten in the form

$$\begin{aligned} A_j U_{j+1}^{k+1} -\left( B_j U_j^{k+1} -E_j V_j^{k+1} \right) +C_j U_{j-1}^{k+1}&= -U_j^k +\Delta t\,fV_\mathrm{g} -s_\mathrm{u} ,\end{aligned}$$

(28a)

$$\begin{aligned} A_j V_{j+1}^{k+1} -\left( B_j V_j^{k+1} +E_j U_j^{k+1} \right) +C_j V_{j-1}^{k+1}&= -V_j^k -\Delta t\, fV_\mathrm{g} -s_\mathrm{v} , \end{aligned}$$

(28b)

where coefficients $A_{j}$, $B_{j}$, and $C_{j}$ are defined by Eq. 20, and

$$\begin{aligned} E_j =f_ \Delta t. \end{aligned}$$

(29)

The above system is equivalent to the matrix equation

$$\begin{aligned} \underline{A}_j \underline{W}_{j+1} -B_j \underline{W}_j +C_j \underline{W}_{j-1} =-\underline{F}_j \end{aligned}$$

(30)

where:

$$\begin{aligned} \underline{A}_{j}&= \left[ \begin{array}{l@{\quad }l} A_{j} &{}0\\ 0 &{}A_{j}\\ \end{array}\right] ,\quad \underline{B}_{j} = \left[ \begin{array}{l@{\quad }l} B_{j}&{} - E_{j} \\ E_{j}&{} B_{j} \\ \end{array} \right] ,\quad \underline{C}_{j} = \left[ \begin{array}{l@{\quad }l} C_{j}&{} 0\\ 0&{} C_{j} \\ \end{array} \right] ,\\ \underline{F}_{j}&= \left[ \begin{array}{l} \underline{U}_{j}^{k} - f\Delta t V_\mathrm{g} + s_\mathrm{u} \\ V_{j}^{k} + f\Delta t U_\mathrm{g} + s_\mathrm{v}\\ \end{array} \right] ,\quad \underline{W}_{j} = \left[ \begin{array}{l} U_{j}^{k + 1} \\ V_{j}^{k + 1}\\ \end{array} \right] \\ \end{aligned}$$

and $j = 2,\ldots , n-1$.

The boundary conditions can be expressed in the general form, which is consistent with Eq. 30

$$\begin{aligned} \underline{A}_{1}\underline{W}_{2} + B_{1}{W}_{1}\! = \underline{F}_{1}\quad \hbox {for}\; i =1,\end{aligned}$$

(31a)

$$\begin{aligned} B_{n}\underline{W}_{n} + C_{n}\underline{W}_{n-1}\! = \underline{F}_{n}\quad \hbox {for}\; i = n. \end{aligned}$$

(31b)

The solution of Eq. 28 is assumed to have the following form:

$$\begin{aligned} \underline{W}_{j-1} =\underline{a}_j \underline{W}_j +\underline{b}_j , \end{aligned}$$

(32)

where $\underline{a}_{j}$ and $\underline{b}_{j}$ are matrices 2x2, and $j = n, {\ldots }, 2$.

Substituting Eq. 32 into Eq. 30, yields:

$$\begin{aligned} \underline{a}_{j+1}&= \left( \underline{B}_j -\underline{C}_j \underline{a}_j \right) ^{-1}\underline{A}_j ,\end{aligned}$$

(33a)

$$\begin{aligned} \underline{b}_{j+1}&= \left( \underline{B}_j -\underline{C}_j \underline{a}_j \right) ^{-1}\left( \underline{F}_j +\underline{C}_j \underline{b}_j \right) , \end{aligned}$$

(33b)

where $i = 2\ldots n-1$. From the boundary conditions we obtain

$$\begin{aligned}&\underline{a}_2 =-\underline{B}_1 ^{-1}\underline{A}_1 ,\end{aligned}$$

(34a)

$$\begin{aligned}&\underline{b}_2 =\underline{B}_1 ^{-1}\underline{F}_1 , \end{aligned}$$

(34b)

and

$$\begin{aligned} \underline{W}_{n} = \left( \underline{B}_{n} + \underline{C}_{n} \underline{a}_{n}\right) ^{-1}\left( \underline{F}_{n} - \underline{C}_{n} \underline{b}_{n}\right) . \end{aligned}$$

(35)

Specifically, when $A_{1} = 0$, $B_{1} = E, F_{1} = 0$, $B_{n} = E$, $C_{n} = - E$ (where E is an identity matrix), then $\underline{a}_2= 0$, $\underline{b}_2= 0$, and $\underline{W}_{n} = (E -\underline{a}_{n})^{-1} (\underline{F}_{n} + \underline{b}_{n})$.

In order to find a solution, the matrices $\underline{a}_j$ and $\underline{b}_j$ were found from Eqs. 33 and 34 within the first “sweep”. Then, the solution was obtained in the second “sweep” from Eqs. 35 and 32.