1 Introduction

In the past few years, flows over inclined terrain have become a focus of intensive research. Recent experiments, such as the MATERHORN-X (Fernando et al. 2015) and SELF-2010 (Nadeau et al. 2013) field campaigns provided new information that encouraged many subsequent studies (e.g., Oldroyd et al. 2014, 2016a; Grachev et al. 2016). One of the important aspects of this research is the turbulent exchange and turbulence structure close to the surface. A recent analysis of observational data by Oldroyd et al. (2016b) has shown the important role of the along-slope heat flux in the buoyant generation or suppression of the turbulent kinetic energy (TKE), thus supplementing earlier work (e.g., Horst and Doran 1988; Denby 1999; Grachev et al. 2016).

The buoyancy term in the TKE budget contains the vertical component of the turbulent heat flux, and may simply be shown to describe the work done by or against the buoyancy force during vertical displacements of particles involved in turbulent motions. In a horizontally homogeneous atmospheric boundary layer (hereafter, the abbreviation ABL will be used, and it will also imply the horizontally homogeneous case), horizontal components of the turbulent heat flux play no role in this energy transformation process. They do not contribute to the flux divergence, even though horizontal turbulent heat-flux components may be several times larger than the vertical component, especially in near-neutral and stable stratification (e.g., Wyngaard et al. 1971). Possibly due to this, horizontal turbulent heat-flux components have previously received relatively little attention. However, over sloping terrain, turbulent heat-flux components are tangent to the surface cross-equipotential surfaces, thereby enabling energy transformation that can be significant considering the magnitude of turbulent heat-flux components. The purpose of the present study is to draw attention to the important aspect of this energy conversion process, i.e, the exchange between the TKE and the turbulent potential energy (TPE). Here, we focus on the direct effects of surface inclination on turbulence, leaving aside the indirect effects related to the formation of the structure of slope flows.

2 The Framework

2.1 The Coordinate System

A derivation of the conservation equations and second-moment turbulence budgets (e.g., Stull 1988) typically involves a horizontal planar Cartesian coordinate system, with one of its axes running vertically, i.e. parallel to the gravity-force vector. However, use of this system to analyze flows over slopes is rather inconvenient, and so hereafter we use a Cartesian coordinate frame (\(x_1, x_2, x_3\)), termed the ‘slope coordinates’, with its \(x_3\) axis perpendicular to the terrain, and the (\(x_1, x_2\)) plane tangent to the slope; the \(x_1\) axis orientation may still be freely chosen, e.g., streamwise, downslope/upslope, or eastward (see discussion in Oldroyd et al. 2016a). There are two basic approaches to express the conservation equations in slope coordinates. One is to transform the equations derived in horizontal planar coordinates using a transformation tensor (e.g., Łobocki 2014, Appendix 2). The other relies on noting that the scalar quantities and their budgets are independent of coordinate system rotations, and that relations between tensors do not depend on the choice of coordinate frame, e.g. the gradient-transfer theory (or, Fick’s law) states that the flux is directed opposite to the gradient vector, regardless of the orientation of coordinates. However, in this latter approach it must be ascertained that (1) vectors having a fixed orientation in space such as the gravity force or the Earth’s angular velocity are properly projected onto the axes of the coordinate frame in its particular orientation, (2) the equations must be used in their general form, without simplifications motivated by anisotropy, horizontal homogeneity, etc., and (3) referring to vertical and horizontal components of vectors requires remapping their coordinates to the horizontal planar system. Having these requirements satisfied, one may freely choose the particular orientation of the coordinate frame, an option that is adopted here. Throughout the text, summation convention is also used.

In the following analyses, we focus only on the part of the equation system governing the turbulent flow over inclined terrain. Specifically, we investigate the buoyant production terms in second-moment budgets, while omitting the Coriolis, pressure, diabatic heating, turbulent and viscous transport terms. As the buoyancy force is oriented vertically, it is necessary to begin with the specification of the vertical at any arbitrary position of the coordinate frame. Let us denote components of the gravity vector by \((g_1, g_2, g_3)\); hereafter, we use \(g_i\) to denote the gravity vector, for brevity. In particular, in a horizontal planar position of the coordinate frame, \(g_i = - \delta _{i3}\, g\), where g is the numeric value of the acceleration due to gravity. Now, at any position of the coordinate frame, the vertical component \(a_z\) of a given vector \(a_i\) can be found by an orthogonal projection onto the direction of the gravity force, so that the transformation rule simply reads

$$\begin{aligned} a_z = - \dfrac{a_i g_i}{g} \, . \end{aligned}$$
(1)

For an easier perception of results presented in the following sections, traditional annotations of axes (x, y, z) and velocity components (u, v, w) are used when the coordinate frame is in its horizontal position.

2.2 Second-Moment Budget Equations

Let us consider the turbulent motion of a Boussinesq fluid. Omitting the Coriolis, pressure, diabatic heating, turbulent transport and viscous transport terms, one can write heat flux, potential temperature variance and TKE budgets as (see e.g., Stull 1988),

$$\begin{aligned} \dfrac{\mathcal {D}\,{\overline{u'_i \theta '}}}{\mathcal {D}t}= & {} - \overline{u'_i u'_k }\,\frac{\partial {{\varTheta }}}{\partial {x_k}} \, - \overline{u'_k \theta '}\, \frac{\partial {U_i}}{\partial {x_k}} - \frac{g_i}{\mathring{{\varTheta }}} \, \overline{\theta '^2}- 2 \, \varepsilon _{u_i \theta } \,, \end{aligned}$$
(2)
$$\begin{aligned} \dfrac{\mathcal {D}\,{\overline{\theta '^2}}}{\mathcal {D}t}= & {} - 2 \, \overline{u'_k \theta '}\, \frac{\partial {{\varTheta }}}{\partial {x_k}} - 2 \, \varepsilon _\theta \,, \end{aligned}$$
(3)
$$\begin{aligned} \dfrac{\mathcal {D}\,{\overline{u'_k u'_k }}}{\mathcal {D}t}= & {} -2 \overline{u'_k u'_i }\, \frac{\partial {U_i}}{\partial {x_k}} - 2 \, \frac{g_k}{\mathring{{\varTheta }}} \, \overline{u'_k \theta '}- 2 \, \varepsilon \,, \end{aligned}$$
(4)

where capital letters are used to denote mean flow variables, primed lower case letters denote turbulent fluctuations, overbars denote Reynolds averaging, and the summation convention is used. \({\varTheta }\) stands for the mean potential temperature and \(\theta '\) for its fluctuation, \(u_i\) for the i-th velocity component, \(\mathring{{\varTheta }}\) is the reference potential temperature as used in the Boussinesq approximation, and \(\varepsilon \), \(\varepsilon _\theta \) and \(\varepsilon _{u_i \theta }\) denote the rate of dissipation of TKE, potential temperature variance and heat-flux components, correspondingly. The \({\frac{\mathcal {D}}{\mathcal {D}t}}=\frac{\partial {}}{\partial {t}} + U_k\frac{\partial {}}{\partial {x_k}}\) symbol designates the material derivative in the mean flow.

Without omission of any tensor components, these equations hold in any orientation of the coordinate frame. Over horizontal terrain though, Eqs. 2–4 can be simplified, since the buoyancy terms \(\displaystyle {\overline{\theta '^2}\, {g_i}/{\mathring{{\varTheta }}}}\) drop out from the horizontal heat-flux budgets; further, when horizontal components of the gradients of mean fields become negligible, scalar products reduce to arithmetic products involving terms that contain vertical gradients only.

3 Turbulent Potential Energy

The budget of the potential temperature variance \(\overline{\theta '^2}\) is generally included in atmospheric turbulence models and its importance has been shown in many studies. Here, we only mention the most important results in the present context. Mellor and Yamada (1974) conducted a systematic scale analysis of second-moment budget equations and found that the first-order approximation should retain TKE and \(\overline{\theta '^2}\) budgets in their full prognostic form while other equations may be simplified to their local production–dissipation balance, algebraic forms (the so-called ‘level-3’ model). This expectation was supported by numerical results of boundary-layer evolution observed during the Wangara experiment (Yamada and Mellor 1975). Subsequently, a simpler ‘level-2.5’ model, using a local production–dissipation balanced \(\overline{\theta '^2}\) budget, has been widely adopted for practical applications. However, in more recent studies (e.g., Nakanishi 2001; Nakanishi and Niino 2006), substantial improvements were reported when using using the full prognostic form of the \(\overline{\theta '^2}\) budget. Further, Denby (1999) showed the importance of \(\overline{\theta '^2}\) transport in katabatic flows, especially in the region of the jet core. An explanation of these observations stems from the algebraic transformation of the level-3 model equations (Nakanishi 2001; Nakanishi and Niino 2006), which shows that the departure of the value of \(\overline{\theta '^2}\) from its local equilibrium value is responsible for the appearance of non-gradient heat transport in the model.

Dalaudier and Sidi (1987) were probably the first to note that \(\overline{\theta '^2}\) is closely linked to the potential energy of fluid particles in a stratified turbulent flow displaced from their equilibrium (with no density departures from the surroundings). The emerging concept of TPE has been explored in subsequent works (e.g., Zilitinkevich et al. 2007). Notably, they introduced a model based on a budget of total turbulent mechanical energy (the sum of TKE and TPE, hereafter TME) and demonstrated its ability to reproduce features of turbulence in a very stable boundary layer. Here, we propose a formulation of TPE suitable for use in arbitrarily-oriented coordinate systems and discuss its participation in the TME budget.

Let us consider a parcel in a stably-stratified fluid that has attained a certain potential temperature departure \(\theta '\) from the surrounding fluid as a result of vertical displacement in a turbulent motion. If such a parcel is denser than the surrounding fluid, it has some potential energy with respect to the state resulting from a descent to a level where its density would be equal to that of its environment. During this motion, said potential energy is transformed to kinetic energy through work done by gravity and buoyancy forces. In an arbitrarily-oriented coordinate frame, an element of this work in Boussinesq fluids can be found as a scalar product of the resultant buoyancy force and the elementary displacement vector \(\,\mathrm {d}x_k\),

$$\begin{aligned} \,\mathrm {d}W_A = g_k \, \frac{\theta '}{\mathring{{\varTheta }}} \, \,\mathrm {d}x_k = \frac{ g_k }{\mathring{{\varTheta }}} \, u_k' \theta ' \,\mathrm {d}t = - \frac{ g }{\mathring{{\varTheta }}} \, w' \theta ' \,\mathrm {d}t \,, \end{aligned}$$
(5)

where in the last form the \(w'\) symbol is used to emphasize the vertical orientation of the \(x_3\)-axis that may be chosen when convenient. Thus, the second term on the right-hand side of (4) expresses the average power (per unit mass) engaged in this process. Note that the scalar product in (5) expresses an orthogonal projection of the fluctuation velocity (and thereby the heat-flux vector) onto the vertical.

For the moment, we focus on a special case of the horizontally-homogeneous ABL. In this case, the \(x_3\)-axis of the coordinate frame is oriented vertically, so that we use (x, y, z) and (u, v, w) annotations for clarity. Here, (5) may be rewritten as

$$\begin{aligned} \,\mathrm {d}W_A = - \frac{ g }{\mathring{{\varTheta }}} \, w' \theta ' \,\mathrm {d}t = - \frac{g}{\mathring{{\varTheta }}} \, \theta ' \, \,\mathrm {d}z = - \frac{g}{\mathring{{\varTheta }}} \,\left( \frac{\partial {{\varTheta }}}{\partial {z}}\right) ^{-1} \, \theta ' \, \,\mathrm {d}{\varTheta }\,. \end{aligned}$$
(6)

Since the disturbance \(\theta '\) results from the considered process, one may take \(\,\mathrm {d}{\varTheta }= - \,\mathrm {d}\theta '\), which, upon substitution into (6), integration over individual particle paths, and ensemble averaging gives

$$\begin{aligned} {\varDelta }W_A = \frac{g}{\mathring{{\varTheta }}} \,\left( \frac{\partial {{\varTheta }}}{\partial {z}}\right) ^{-1} \, {\left( \frac{\overline{\theta '^2}}{2}\right) } , \end{aligned}$$
(7)

which leads to the definition of the turbulent potential energy

$$\begin{aligned} \mathscr {E_P}= \frac{g}{\mathring{{\varTheta }}} \,\left( \frac{\partial {{\varTheta }}}{\partial {z}}\right) ^{-1} \, {\left( \frac{\overline{\theta '^2}}{2}\right) } \,, \end{aligned}$$
(8)

where the reference state (zero energy) is the lack of \(\theta '\) fluctuations, that is, \(\overline{\theta '^2}=0\). For convenience, let us denote

$$\begin{aligned} \gamma = \frac{g}{\mathring{{\varTheta }}} \,\left( \frac{\partial {{\varTheta }}}{\partial {z}}\right) ^{-1}. \end{aligned}$$
(9)

The above interpretation assumes that the vertical gradient of potential temperature is positive. At the neutral equilibrium, Eq. 8 would become singular; in contrast, the use of the \(\overline{\theta '^2}\) budget does not suffer from singularity issues nor problems with the interpretation of negative TPE and its dissipation in unstable conditions. However, the TPE concept provides an illuminating physical interpretation of \(\overline{\theta '^2}\).

In a thermally-stratified ABL, due to the absence of horizontal components of \(\nabla {\varTheta }\), multiplication of (3) by \(\gamma \) leads to a TPE budget in which the production term turns out to be identical to the buoyant production term in the TKE budget, just with an opposite sign. Thus, these terms cancel out when the TKE and TPE budgets are combined. However, when horizontal components of \(\nabla {\varTheta }\) exist, expanding the production term in (3) yields expressions that have no counterpart in the TKE budget. In other words, while the flux-gradient scalar product in (3) reflects an orthogonal projection of \(\nabla {\varTheta }\) onto the direction of the turbulent heat-flux vector, the scalar product of the body force and \(\nabla {\varTheta }\) in (4) involves the vertical component of \(\nabla {\varTheta }\) only. In a more general situation where \(\nabla {\varTheta }\) may be inclined at a certain angle (for example, in katabatic flows), these additional terms require consideration.

The classic parcel method used in calculating (7) does not take into account compensating motions in the environment of the vertically displaced parcel. As we shall see, in vertically-stable baroclinic situations these motions become the virtue of the potential energy release, hence a more general approach is necessary. An adequate candidate is the method used in large-scale geophysical fluid dynamics to explain baroclinic instability (e.g., Vallis 2006, Sect. 6.4.1).

Fig. 1
figure 1

A schematic picture of energy conversion paths (in a vertical cross-section) under statically stable conditions, in the presence of a horizontal temperature gradient. The X-axis is aligned with the horizontal component of \(\nabla {\varTheta }\) and the Z-axis points vertically upwards; dashed coloured lines are isentropes (red symbolizes the highest value). Capital letters mark positions while lowercase letters mark moving parcels. Arrows symbolize hypothetical paths of interchanging parcels’ positions. Left a transposition of two parcels; right the position of parcels after a cyclic shift by one section

Full size image

Let us examine potential energy changes resulting from transpositions, i.e., interchanges of positions taking place between pairs of air parcels, in a baroclinic situation depicted in the left panel of Fig. 1. Here, isentropes are inclined at a certain angle \(\varphi \) with respect to the horizontal. This temperature pattern is frequently assumed (e.g., Horst and Doran 1988; Denby 1999; Oldroyd et al. 2016b) over slopes. Since an air parcel moving along an isentrope of the same \({\varTheta }\) value is not buoyant, the work performed by the resultant buoyancy and gravity forces is null. The same applies to any interchange of two air parcels (e.g., b and d in Fig. 1) along an isentrope. However, a transposition along a gentler path (e.g., b with c) involves potential energy release (hence, it can happen spontaneously) and causes buoyancy forces to appear that may act to increase the initial disturbance (hence, the state is unstable despite the positive vertical component of \(\nabla {\varTheta }\)). While this mechanism is well-known in large-scale dynamics, its role in small-scale motions such as turbulent vortices is yet to be established. Nevertheless, it constitutes a possible way for potential energy release under apparently stable conditions (according to the convective stability criterion) that may contribute to the kinetic energy of turbulent vortices (excited, e.g., by shear instability); the required condition is the presence of a horizontal component of \(\nabla {\varTheta }\).

Following Vallis (2006, Eq. 6.61), the change of potential energy of the pair of parcels b and c resulting from a transposition, may be calculated as (per unit volume)

$$\begin{aligned} g \left( \rho _b z_1 + \rho _c z_2 - \rho _b z_2 - \rho _c z_1\right) = g \left( \rho _b-\rho _c \right) \left( z_1-z_2\right) =g \,\delta \rho \, \delta z \,. \end{aligned}$$
(10)

For a shallow flow, where the Boussinesq approximation is applicable, this expression can be rewritten as

$$\begin{aligned} g \, \delta \rho \, \delta z = g \left( \delta x \, \frac{\partial {\rho }}{\partial {x}} + \eta \,\delta x \, \frac{\partial {\rho }}{\partial {z}}\right) \eta \, \delta x = - \mathring{\rho }\, \frac{g}{\mathring{{\varTheta }}} \frac{\partial {{\varTheta }}}{\partial {z}} \, \delta x^2 \, \eta \left( 1 - \frac{\eta }{\phi } \right) \,, \end{aligned}$$
(11)

where \(\eta = \,\mathrm {d}z / \,\mathrm {d}x\) is the inclination of the transposition path and \(\phi = \delta z / \delta x\) is the inclination of isentropes. The potential energy release is null when \(\eta =0\) or when \(\eta =\phi \), positive when \(0<\eta <\phi \), with a maximum at \(\eta =\phi /2\), and negative for \(\eta >\phi \).

Now, let us return to energy budgets. Since, according to Eq. 1,

$$\begin{aligned} \displaystyle {\frac{\partial {{\varTheta }}}{\partial {z}}= - \frac{g_m}{g} \frac{\partial {{\varTheta }}}{\partial {x_m}} } \,, \end{aligned}$$
(12)

one may obtain the TPE budget equation from (3) as

$$\begin{aligned} \dfrac{\mathcal {D}\,{\mathscr {E_P}}}{\mathcal {D}t} = \frac{g}{\mathring{{\varTheta }}} \left( \frac{g_m}{g} \frac{\partial {{\varTheta }}}{\partial {x_m}} \right) ^{-1} \left( \, \overline{u'_k \theta '}\, \frac{\partial {{\varTheta }}}{\partial {x_k}} + \varepsilon _\theta \right) \,, \end{aligned}$$
(13)

where in an arbitrary orientation of the coordinate frame TPE can be calculated as

$$\begin{aligned} \mathscr {E_P}= \gamma \, \frac{\overline{\theta '^2}}{2} \,, \end{aligned}$$
(14)

and \(\gamma \) is now rewritten as

$$\begin{aligned} \gamma = - \frac{g^2}{\mathring{{\varTheta }}} \left( g_m \frac{\partial {{\varTheta }}}{\partial {x_m}} \right) ^{-1} \,. \end{aligned}$$
(15)

These are the forms that should be used in an arbitrarily-oriented slope coordinate frame, with an arbitrary orientation of the potential temperature gradient.

To explain the structure of the production term in (13), let us choose the horizontal planar orientation of the coordinate frame, with the x-axis pointing in the direction of the horizontal temperature gradient. Here, this term can be written as

$$\begin{aligned} \frac{g}{\mathring{{\varTheta }}} \left( \overline{u'\theta '}\, \frac{ {\partial {\varTheta }}/{\partial x} }{{\partial {\varTheta }}/{\partial z}} + \overline{w'\theta '}\right) = \frac{g}{\mathring{{\varTheta }}} \left( \phi \, \overline{u'\theta '}\, + \overline{w'\theta '}\right) \,, \end{aligned}$$
(16)

since \({\partial {\varTheta }}/{\partial y}\) is null in this orientation, the ratio of gradients equals the inclination of isentropic surfaces \(\phi ={ \delta z }/{ \delta x }\), and \(\gamma \) reduces to \((g/\mathring{{\varTheta }}) \, {\partial {\varTheta }}/{\partial z}\). Geometrically, the first term in brackets represents a parallel oblique projection of the horizontal heat-flux vector onto the vertical, along isentropes as projectors. In cyclic shifts, \(u'\) and \(w'\) have opposite signs, hence in situations such as that presented in Fig. 1, compensation will take place between terms in (16). For \(-w'/u'=\phi \) (parcel moving along an isentrope), these two terms completely cancel each other. However, when \(-w'/u'=\eta <\phi \), the compensation is only partial, and a surplus of \(-\phi \,\overline{u'\theta '}\) over \(\overline{w'\theta '}\) appears, consistent with (11).

According to (5), the term \((g/\mathring{{\varTheta }}) \, \overline{w'\theta '}\) represents the mechanical work done during vertical displacement of a parcel. Hence, the remaining term \( (g/\mathring{{\varTheta }})\, \phi \, \overline{u'\theta '}\) should also be interpreted as work, evidently performed during horizontal displacements. Let us consider a single-step circular shift in a conservative field of forces, within a hypothetical closed chain of three parcels: a, b, and d (Fig. 1, right panel), initially located at A, B, and D. The change of the potential energy of this triplet is, again

$$\begin{aligned} g \left( \rho _a z_1 + \rho _b z_1 + \rho _d z_3 - \rho _a z_1 - \rho _b z_3 - \rho _d z_1\right) =g \left( \rho _b-\rho _d\right) \left( z_1-z_3\right) , \end{aligned}$$
(17)

which can be rewritten in terms of \({\varTheta }\) as in (11). As already explained, in this special case the work done on a parcel b during its way from B to D is null. Hence, the mechanical work \((g/\mathring{{\varTheta }}) \, \overline{w'\theta '}\) done on parcel d must be fully compensated by the work \( (g/\mathring{{\varTheta }})\, \phi \, \overline{u'\theta '}\) performed by parcel a while moving from A to B.Footnote 1 Obviously, when B and D lie on the same isentrope, there is no net change in potential energy of this system; however, should the top point lie below the isentrope running through B (as does the point C in the left panel), with \(\rho _b\ne \rho _c\) and \(z_1 \ne z_2\) (in Eq. 17, \(\rho _d\) should be replaced with \(\rho _c\) and \(z_3\) with \(z_2\) in this case), a change will occur. In particular, when the interchange process is shallow enough (e.g. the ABC path), \(-w'/u'<\phi \), and the \(\phi \,\overline{u'\theta '}\) term in (16) prevails.

As the continuity of fluid requires that any parcel leaving a given place must be replaced by another parcel of equal volume, and that it displaces other parcel(s) during its motion, a closed chain of such displacements must occur in any contiguous area of a finite size. Further, as follows from (17), adding extra elements to such a chain does not change the increase/decrease of the total potential energy of the chain elements. Therefore, it is sufficient to consider a cyclic shift in a closed three-element chain presented in Fig. 1. Here, continuity requires that \(w'_b=-w'_d\) and \(u'_a=-u'_b\) so that \(w'_b/u'_a=w'_d/u'_a=w'/u'=\eta \). Further, it is also evident that when \(\eta =\phi \), then \(\theta '_d=-\theta '_a\) and \(\theta '_b=0\); hence \(-\phi u'\,\theta '=w'\theta \), in agreement with the previously presented thermodynamic argument. But, when the motion is shallower (e.g., when parcel b reaches C instead of D, left panel), \(\theta '_c<-\theta '_a\), an imbalance in (16) appears, according to (17). Hence, the term \( (g/\mathring{{\varTheta }})\, \phi \, \overline{u'\theta '}\) may be identified as work performed on parcel a by the horizontal pressure-gradient force arising from uneven hydrostatic load. As the potential energy gained in this process may be fully released during a subsequent descent to the same isentrope, this work is also equal to the energy necessary for a vertical displacement of parcel d, to the distance at which the potential temperature fluctuations \(\theta '_d\) and \(\theta '_a\) become compensated. It is noteworthy that while the second term in (16) cancels out the buoyant production term in the TKE budget, the first one does not. Hence, the source term in the TME budget is the difference reflecting the loss of potential energy of the mean field resulting from parcel transpositions.

While this idealized model is helpful in understanding the supply of energy, other processes, such as mixing, conversion to kinetic energy, and dissipation must also occur in order to maintain a constant energy flow. While they may occur only at some stages of the motion, or act continuously, the potential energy amount available for transformation is the same. Further, as eddies of different orientations, sizes and aspect ratios exist in a turbulent flow, this mechanism may supply energy to a certain part of these motions, thus weakening the damping effect of static stability. One may speculate whether this mechanism might possibly lead to relaxing the criteria for turbulence decay, such as the critical Richardson number.

To explain the nature of the work done during horizontal displacements, as the pressure terms are eliminated during the derivation of turbulent energy budgets, it is necessary to recall the momentum budget. Let us consider the horizontal pressure gradient arising from differences in hydrostatic load due to the presence of horizontal temperature and density gradients. Under the Boussinesq approximation, the thermal wind relationship in height coordinates (Vallis 2006, Eqs. 2.205 and 2.190) yields the rate of change of the pressure gradient force with height,

$$\begin{aligned} f\frac{\partial {v_g}}{\partial {z}} = \frac{\partial {}}{\partial {z}} \frac{\partial {}}{\partial {x}}\left( \frac{\delta p}{\rho _0}\right) = \frac{g}{\mathring{{\varTheta }}} \frac{\partial {{\varTheta }}}{\partial {x}} \,, \end{aligned}$$
(18)

where f is the Coriolis parameter and \(v_g\) is the y-component of the geostrophic velocity. Assuming \(\nabla {\varTheta }\) to be constant within a certain height range \(\delta z\), the resulting difference of horizontal pressure gradient across this layer would be

$$\begin{aligned} \int _z^{z+\delta z} \frac{g}{\mathring{{\varTheta }}} \frac{\partial {{\varTheta }}}{\partial {x}} = \frac{g}{\mathring{{\varTheta }}} \frac{\partial {{\varTheta }}}{\partial {x}} \delta z\,. \end{aligned}$$
(19)

Further, during an infinitesimal horizontal displacement of a parcel at a distance of \(\delta x = u' \delta t\), the pressure-gradient force performs work equal to the product of the right-hand side of (19) and \(u' \, \delta t\). Then, one can substitute \(\delta z = \phi \, \delta x\), using the isentropes’ inclination \(\phi \). As the parcel’s motion is considered adiabatic, the potential temperature fluctuation resulting from this displacement may be found as \(\theta '=\partial {\varTheta }/ \partial x \,\delta x\). Now, taking the Reynolds average of the result, we arrive at the first term in (16).

To verify the correctness of geometric transformations, and to compare with the ABL situation, let us consider a special case where \(\nabla {\varTheta }\) is oriented vertically, while the coordinate frame is used in an arbitrary orientation. In this situation, the production term in Eq. 3 reduces to

$$\begin{aligned} \overline{u'_j \theta '}\, \frac{\partial {{\varTheta }}}{\partial {x_j}} = \left( \frac{g_k}{g} \, \overline{u'_k \theta '}\ \right) \, \left( \frac{g_m}{g} \, \frac{\partial {{\varTheta }}}{\partial {x_m}} \right) \,, \end{aligned}$$
(20)

i.e. to a product of an orthogonal projection of the turbulent heat flux onto the vertical and the vertical component of \(\nabla {\varTheta }\), as scalar products are invariant in coordinate system rotations. The TPE budget (13) assumes the form

$$\begin{aligned} \dfrac{\mathcal {D}\,{\mathscr {E_P}}}{\mathcal {D}t} = \frac{g_k}{\mathring{{\varTheta }}} \, \overline{u'_k \theta '}\, - \gamma \, \varepsilon _\theta \,. \end{aligned}$$
(21)

Noteworthy is that, in an arbitrary orientation of the coordinate system, all the components of the turbulent heat flux contribute to the TPE budget, as long as the corresponding gravity force components are not null. Should the coordinate frame be brought to the horizontal planar position, the \(g_1\) and \(g_2\) components would become null. As we see now, only the vertical component of the turbulent heat flux, \(\overline{w'\theta '}\), directly contributes to TPE, according to the previous discussion. Further, the buoyancy work term is identical to that in the TKE budget but has an opposite sign. Hence, it cancels out upon combination of TKE and TPE budgets into the TME budget

$$\begin{aligned} \dfrac{\mathcal {D}\,{(\mathscr {E_K}+\mathscr {E_P})}}{\mathcal {D}t} = - \overline{u'_k u'_i }\, \frac{\partial {U_i}}{\partial {x_k}} - \varepsilon - \gamma \, \varepsilon _\theta . \end{aligned}$$
(22)

However, in the presence of non-zero horizontal components of \(\nabla {\varTheta }\) (in particular, when the isentropes run parallel to the inclined surface, as might be expected in slope flows close to the ground), additional terms arise in the expansion of sums in Eq. 20 as a result of the work done during horizontal displacements by the pressure-gradient force (as discussed above), and they must be included in the TME budget.

4 Pure Slope Flows

To facilitate a qualitative analysis, we consider a special case of flow directed along the slope, without a cross-slope component. Let the \(x_1\) axis be oriented streamwise (and along the slope); in this case, \(g_i = (-g \sin \alpha ,\,0,\,-g \cos \alpha )\), where \(\alpha \) is the slope inclination angle (here, taken positive for upslope flow and upslope \(x_1\)-axis orientation). Further, we adopt a frequently made assumption (e.g., Horst and Doran 1988; Denby 1999; Oldroyd et al. 2016b) that the potential temperature and wind-speed gradients are perpendicular to the slope. With these settings, Eqs. 13 and 4 may be rewritten as

$$\begin{aligned} \dfrac{\mathcal {D}\,{\mathscr {E_P}}}{\mathcal {D}t}&= - \gamma \, \overline{u'_3 \theta '}\, \frac{\partial {{\varTheta }}}{\partial {x_3}} - \gamma \, \varepsilon _\theta \,, \end{aligned}$$
(23)
$$\begin{aligned} \dfrac{\mathcal {D}\,{\mathscr {E_K}}}{\mathcal {D}t}&= - \overline{u_1' u_3'} \, \frac{\partial {U_1}}{\partial {x_3}} + \frac{g }{\mathring{{\varTheta }}} \, \overline{u_1' \theta '} \sin \alpha + \frac{g }{\mathring{{\varTheta }}} \, \overline{u_3' \theta '} \cos \alpha - \, \varepsilon . \end{aligned}$$
(24)

In the horizontal planar orientation of the coordinate system, the \(\overline{\theta '^2}\) production term in Eq. 23 expands to two elements

$$\begin{aligned} \overline{w'\theta '}\, \frac{\partial {{\varTheta }}}{\partial {z}}= & {} \overline{u'_3 \theta '}\, \frac{\partial {{\varTheta }}}{\partial {x_3}} \cos ^2 \alpha , \end{aligned}$$
(25a)
$$\begin{aligned} \overline{u'\theta '}\, \frac{\partial {{\varTheta }}}{\partial {x}}= & {} \overline{u'_3 \theta '}\, \frac{\partial {{\varTheta }}}{\partial {x_3}} \sin ^2 \alpha \,. \end{aligned}$$
(25b)

Hence (also due to the scalar invariance)

$$\begin{aligned} \overline{u'_3 \theta '}\, \frac{\partial {{\varTheta }}}{\partial {x_3}} = \overline{w'\theta '}\frac{\partial {{\varTheta }}}{\partial {z}}+ \overline{u'\theta '}\frac{\partial {{\varTheta }}}{\partial {x}}\,, \end{aligned}$$
(26)

which, upon multiplication by \(\gamma \), yields the production term in (23) expressed as in (16). As discussed earlier, this term reflects the part of the work done by the pressure-gradient force during horizontal displacement that is not compensated for by the work used for vertical displacement in ‘shallow’ motions, where the effective transposition paths have smaller inclinations than isentropes (and the slope). As the extent of such motions represents an additional degree of freedom, constraints of energy transformations related to the slope inclination should be treated as asymptotic rather than actual. Perhaps, this result may explain the spread of experimental data as shown in Oldroyd et al. (2016b), their Fig. 2.

As a final remark of this section, the often-made assumption of isentropes parallel to the terrain in katabatic flows has not been sufficiently explored and deserves further research. Changes in isentropic inclination may reflect uneven cooling along the slope, the shape of gravity currents, terrain irregularities, temporal changes, and so on. As the mathematical analysis presented here does not rest on such a limiting assumption, it may provide a framework for further studies of such effects.

5 Turbulent Heat Flux

The discussion in the previous sections relates to mechanisms of energy exchange between the mean flow and turbulence, and the role of individual heat-flux components in these processes. Therefore, it is interesting to inspect turbulent heat-flux budget intricacies. To simplify the analysis, let us continue the investigation of the special case of a pure slope flow, introduced in the previous section. The turbulent heat-flux budget equation (Eq. 2) may be rewritten as

$$\begin{aligned} \dfrac{\mathcal {D}\,{\overline{u_1' \theta '}}}{\mathcal {D}t}&= - \overline{u_1' u_3'} \, \frac{\partial {{\varTheta }}}{\partial {x_3}} \, - \overline{u_3' \theta '} \, \frac{\partial {U_1}}{\partial {x_3}} + \frac{g}{\mathring{{\varTheta }}} \, \overline{\theta '^2}\, \sin \alpha - 2 \, \varepsilon _{u_1 \theta } \,, \end{aligned}$$
(27)
$$\begin{aligned} \dfrac{\mathcal {D}\,{\overline{u_3' \theta '}}}{\mathcal {D}t}&= - \overline{u_3'^{\,2}} \, \frac{\partial {{\varTheta }}}{\partial {x_3}} \, + \frac{g}{\mathring{{\varTheta }}} \, \overline{\theta '^2}\, \cos \alpha - 2 \, \varepsilon _{u_3 \theta } \,. \end{aligned}$$
(28)

The first two terms on the right-hand side of Eq. 27 describe contributions from shear and static stability in the presence of turbulence (e.g., Wyngaard et al. 1971). It is worth noting that these terms do not contain gravity; therefore, their interpretation should be similar to the case of the ABL. Figure 2 schematically displays situations where the potential temperature increases with the distance from the surface, the turbulent heat flux is directed towards the ground, and the wind speed either increases or decreases with height (e.g., above the jet core). As can be seen, the slope-tangent turbulent heat-flux component arises as a consequence of correlations between slope-normal and streamwise velocity components, when gradients of mean flow values are present. This schematic explanation may apply to different situations identified in slope flows, such as those discussed by, e.g., Horst and Doran (1988), Grachev et al. (2016), Oldroyd et al. (2016b).

Fig. 2
figure 2

A schematic explanation of the horizontal turbulent heat flux under stable conditions, with wind speed increasing (left panel) or decreasing (right panel) with height. The slant curve represents the wind-velocity profile, and the colour symbolizes potential temperature distribution (blue means colder). As the parcel rise is accompanied by both horizontal velocity and potential temperature fluctuations, their coincidence results in a non-zero covariance \(\overline{u'\theta '}\). Note that this explanation remains valid for normal and tangent components in a pure slope flow (imagine a rotation of the picture). However, distortions may appear when the potential temperature gradient inclines with respect to the surface

Full size image

In a horizontal planar coordinate frame, the term containing \(\overline{\theta '^2}\) in Eq. 27, related to TPE by Eqs. 9 and 14, affects only the vertical turbulent heat-flux component. However, in an arbitrary frame orientation it also influences along-slope and cross-slope turbulent heat-flux components. As demonstrated in earlier studies (e.g., Nakanishi and Niino 2006), this term may reflect non-local (non-gradient) contributions to the turbulent heat flux. In the context of katabatic flows, the importance of \(\overline{\overline{\theta '^2}}\) transport, especially near the core of the jet, was shown by Denby (1999).

While TPE plays a role in turbulent heat-flux formation, in turn, this flux controls the TPE production by Eq. 21, as discussed in Sect. 3. This feedback mechanism deserves some attention. Let us consider an equation system consisting of Eqs.  3 and 28 with their dissipation and gradient transport terms omitted, within the ABL, viz.

$$\begin{aligned} \dfrac{\mathcal {D}\,{\overline{w'\theta '}}}{\mathcal {D}t}&= \frac{g}{\mathring{{\varTheta }}} \, \overline{\theta '^2}\, \cos \alpha \,, \end{aligned}$$
(29)
$$\begin{aligned} \dfrac{\mathcal {D}\,{\overline{\theta '^2}}}{\mathcal {D}t}&= - 2 \, \overline{w'\theta '}\, \frac{\partial {{\varTheta }}}{\partial {z}}\,. \end{aligned}$$
(30)

By differentiating Eq. 30 and substituting the derivative of heat flux from Eq. 29, one obtainsFootnote 2

$$\begin{aligned} {\dfrac{\mathcal {D}^2\,}{\mathcal {D}t^2}} \, {\overline{\theta '^2}} = - 2 \, N_{BV}^2 \cos \alpha ~ \overline{\theta '^2}\,, \end{aligned}$$
(31)

where \(N_{BV}=(g/\mathring{{\varTheta }}\partial \theta / \partial z)^{1/2}\) is the Brunt–Väisäla frequency. This equation suggests the presence of an oscillatory mode in the evolution of turbulence. However, as the variances cannot become negative, the production term containing potential temperature variance in Eq. 2 can only cause growth of the vertical component of heat flux, not a decrease. Under stable stratification, this requires reducing the downward heat flux, and a consequential reduction of the TKE destruction rate by buoyancy. It is worth noting though that Eq. 31 predicts a finite time period of \(\overline{\theta '^2}\) destruction, dictated by the Brunt–Väisäla frequency rather than an exponential decay.

On the other hand, according to Eq. 30, when the vertical heat flux and the vertical potential temperature gradient have opposite signs, \({\overline{\theta '^2}}\) should increase, thereby counteracting the increase of the magnitude of the heat flux. The further evolution of this highly idealized process likely differs from typical scenarios, as the above considerations exclude at least two capital factors, shear production and dissipation.Footnote 3 Nevertheless, one could expect conversions between TPE and TKE characterized by the Brunt–Väisäla frequency to be detectable in stably stratified flows.

Notwithstanding the above-discussed limitation, TKE can increase at a cost of TPE under statically-stable equilibrium in regions where countergradient heat flux is present. In such situations, TPE (or, \({\overline{\theta '^2}}\)) is directly consumed by the upward heat flux, which then becomes a source rather than a sink in the TKE budget.

The presence of the Brunt–Väisäla frequency and the restoring role of the buoyancy force may tempt us to associate the mechanism described above with pulsations occurring in katabatic winds (e.g., McNider 1982; Princevac et al. 2008), gravity waves associated with slope winds (e.g., Chemel et al. 2009) and intermittency of turbulence caused by breaking gravity waves (e.g., Staquet 2004). Apparently, these are separate mechanisms that may, but need not necessarily, coexist and affect one another, e.g., by driving changes of shear and stress. The possible feedback deserves attention in further studies.

6 Conclusions

The analysis presented in Sect. 3 exhibits the pathways of potential energy exchange between the mean flow and the turbulence when the potential–temperature-gradient vector, \(\nabla {\varTheta }\), deflects from the vertical. This analysis employs the stability criterion based on potential energy changes resulting from transpositions of pairs of air parcels, instead of the traditional convectional stability concepts based on the parcel method. This method may also be applicable in other situations where baroclinic effects are important.

While many studies of slope flows assume a certain orientation of the \(\nabla {\varTheta }\) vector, a more general framework of arbitrary slope orientation, arbitrary wind direction and arbitrary inclination of the \(\nabla {\varTheta }\) is entered, using a generalized formulation of the TPE concept. Examination of the TPE budget shows that the energy exchange between the potential energy of the mean flow and the TME is controlled by the horizontal component of turbulent heat flux and the inclination of the ambient \(\nabla {\varTheta }\). Consistent thermodynamic, kinematic and dynamic explanations of this phenomenon were given. As the revealed mechanism bears remarkable similarity to a synoptic-scale baroclinic instability, it seems justified to identify the discussed term in the TPE budget as ‘baroclinic generation’.

The general discussion of Sect. 3 is illustrated with the example of a special case, a pure slope flow, in Sects. 4 and 5. Equation 28 specifies the relation between the \(\overline{\theta '^2}\) production terms in slope and horizontal planar coordinate system, exhibiting the contribution from the horizontal turbulent heat-flux component and showing its dependence on the inclination of \(\nabla {\varTheta }\). This result may reveal a possible explanation for the data spread reported in, e.g., Oldroyd et al. (2016b, their Fig. 2) when slope angle was used as a primary controlling parameter without considering the \(\nabla {\varTheta }\) inclination.

As a side note, it is shown that the term containing the potential temperature variance in the turbulent heat-flux budget may act as a restoring mechanism in the flux-energy feedback, with a time scale characterized by the Brunt–Väisäla frequency. While a partial explanation of this mechanism has been given, it may deserve attention in future studies.