# Revisiting and revising Tatarskii’s formulation for the temperature
structure parameter ($C_{T}^{2}$) in atmospheric flows

Sukanta Basu<EMAIL_ADDRESS>Faculty of Civil Engineering and
Geosciences, Delft University of Technology, Delft, the Netherlands  Albert A.
M. Holtslag<EMAIL_ADDRESS>Meteorology and Air Quality, Wageningen
University, Wageningen, the Netherlands

###### Abstract

In this paper, we revisit a well-known formulation of temperature structure
parameter ($C_{T}^{2}$), originally proposed by V. I. Tatarskii. We point out
its limitations and propose a revised formulation based on turbulence variance
and flux budget equations. Our formulation includes a novel physically-based
outer length scale which can be estimated from routine meteorological data.

Bolgiano scaling; Buoyancy-range; Inertial-range; Optical turbulence; Outer
length scale

##  I Introduction

According to the hypothesis by Kolmogorov-Obukhov-Corrsin (KOC; Corrsin, 1951;
Kolmogorov, 1941; Obukhov, 1949) within the inertial-range, the second-order
structure function ($S_{2\theta}^{KOC}$) of potential temperature ($\theta$)
in the vertical direction ($z$) should behave as follows:

$S_{2\theta}^{KOC}=\langle\left(\Delta\theta\right)^{2}\rangle\sim\left(\overline{\varepsilon}\right)^{-1/3}\overline{\chi}_{\theta}\left(\Delta
z\right)^{2/3}.$ (1)

Here $\Delta z$ is a separation distance that varies within the inertial range
$l_{0}\ll\Delta z\ll\Lambda_{0}$. The inner and outer scales of turbulence are
denoted by $l_{0}$ and $\Lambda_{0}$, respectively. The angular bracket and
overlines denote ensemble averaging. The dissipation rates of turbulent
kinetic energy (TKE; $\overline{e}$) and variance of temperature
($\sigma_{\theta}^{2}$) are denoted by $\overline{\varepsilon}$ and
$\overline{\chi}_{\theta}$, respectively.

In the literature, Eq. (1) is commonly re-written as follows:

$S_{2\theta}^{KOC}=C_{T}^{2}\left(\Delta z\right)^{2/3},$ (2a) where
$C_{T}^{2}$ is the so-called temperature structure parameter which is commonly
expressed as Wyngaard et al. (1971):
$C_{T}^{2}=\left(\frac{c}{2}\right)~{}\left(\overline{\varepsilon}\right)^{-1/3}\overline{\chi}_{\theta}=c\left(\overline{\varepsilon}\right)^{-1/3}\overline{N}_{\theta}.$
(2b)

The variable $\overline{N}_{\theta}$ is simply one half of
$\overline{\chi}_{\theta}$ and has been used instead of
$\overline{\chi}_{\theta}$ in a number of publications (including by Tatarskii
Tatarskii (1971)). The proportionality constant $c$ is typically assumed to be
in the range of 2.8–3.2.

Direct measurement of $\overline{\varepsilon}$, $\overline{\chi}_{\theta}$
(or, $\overline{N}_{\theta}$), and $C_{T}^{2}$ is difficult in the atmosphere
as it involves computations of fine-scale velocity and temperature gradients
with spatial resolutions of a few mm (i.e., on the order of $l_{0}$). As a
viable alternative, fifty years ago, Tatarskii Tatarskii (1971) formulated the
following equation:

$C_{T}^{2}=\left(\frac{c}{Pr_{t}}\right)L_{0}^{4/3}\Gamma^{2},$ (3)

where $\Gamma$ is the gradient of mean potential temperature (i.e.,
$\partial\overline{\theta}/\partial z$). $L_{0}$ is a characteristic length
scale; its relationship with $\Lambda_{0}$ will be discussed later. $Pr_{t}$
is known as the turbulent Prandtl number. For simplicity, several studies
(e.g., Beland and Brown, 1988; VanZandt et al., 1978; Walters and Kunkel,
1981) in the past have (incorrectly) assumed $Pr_{t}$ to be equal to unity.

According to Eq. (3), if $L_{0}$ and $\Gamma$ can be reliably measured,
estimated or prescribed, then $C_{T}^{2}$ values can be computed directly.
Since it is relatively straightforward to measure or estimate $\Gamma$ in the
atmosphere, over the past decades, several studies focused on the estimation
of $L_{0}$. Some of these studies (e.g., Coulman et al., 1988; Dewan et al.,
1993; VanZandt et al., 1978) proposed empirical regression equations. Others
Basu (2015); Wu et al. (2020) recommended usage of certain well-known length
scales (e.g., Thorpe scale, Ellison scale) as surrogates of $L_{0}$.

The purpose of the present study is two-fold. First, we revisit the derivation
of Eq. (3) following the footsteps of Tatarskii Tatarskii (1971).
Unfortunately, his original derivation was somewhat cryptic and included
various implicit assumptions. We have attempted to elucidate on the derivation
and the assumptions to the best of our abilities. In addition, we have made a
sincere effort to point out some fundamental limitations of Tatarskii’s
assumptions in the context of atmospheric flows. Next, by relaxing some of
these assumptions, we have derived an alternative to Eq. (3) from fundamental
budget equations of stably stratified flows. In our formulation, $L_{0}$ is
directly related to common meteorological variables (e.g., variance of
potential temperature) and does not require any ad-hoc prescription.

The structure of the paper is as follows. In Section II, we delve into the
original derivation of Eq. (3). Its association with turbulence scaling laws
in the buoyancy-range is explored in Section III. Our proposed formulation is
documented in Section IV. The limitations of both Tatarskii’s and our proposed
formulations for convective mixed layer are discussed in Section V. The
findings of this study are summarized in Section VI. Finally, in Appendix 1,
we derive formal relationships among certain length scales.

##  II Derivation of Tatarskii’s $C_{T}^{2}$ Formulation

Tatarskii utilized both heuristic arguments and turbulence variance budget
equations for the derivation of Eq. (3). They are documented in sections 14
and 15 of Tatarskii (1971). At the outset, it is important to note that these
derivations do not account for the buoyancy effects (discussed below).
However, in section 17 of Tatarskii (1971), buoyancy effects are mentioned in
a different context; there, Tatarskii derived a diagnostic relationship
between $L_{0}$ and $\Lambda_{0}$ following a formalism by Bolgiano Bolgiano
Jr. (1959, 1962) and Obukhov Obukhov (1959). In this paper, we elaborate on
all these derivations albeit with additional clarifications. Furthermore, we
have slightly changed some of the notations used by Tatarskii to be consistent
with the contemporary literature and also with the rest of our manuscript. For
better logical reasoning, we have also changed the sequential ordering of
equations used by Tatarskii.

###  II.1 Heuristic Arguments

The following text has been taken verbatim from Section 14 (page 73) of
Tatarskii (1971):

> “A gradient of mean temperatures will result in a systematic temperature
> difference between any two points at different heights. This temperature
> difference $\Delta\theta$ is approximately given by
> $\Delta\theta\approx\Gamma\Delta z$, and its square is
> $\left(\Delta\theta\right)^{2}\approx\Gamma^{2}\left(\Delta z\right)^{2}$.
> There is also a random temperature difference between these points, whose
> mean square value is $C_{T}^{2}\left(\Delta z\right)^{2/3}$. For small
> $\Delta z$, $C_{T}^{2}\left(\Delta z\right)^{2/3}$ is much greater than the
> gradient term $\Gamma^{2}\left(\Delta z\right)^{2}$ (i.e., the random
> temperature differences are much greater than the systematic ones). There is
> however a certain $\Delta z_{0}$ when the two factors become comparable, and
> for $\Delta z>\Delta z_{0}$ the mean temperature difference is greater than
> the random difference. This $\Delta z_{0}$ is interpreted as the vertical
> mixing scale. Clearly the “2/3 law” is applicable only over distances
> $\Delta z$ not greater than this mixing scale. Therefore $\Delta z_{0}$ may
> be taken equal to the outer scale of turbulence.”

Essentially, Tatarskii suggested equating the inertial-range term (‘random’)
with a larger-scale term (‘systematic’) as follows:

$C_{T}^{2}\left(\Delta z_{0}\right)^{2/3}=\Gamma^{2}\left(\Delta
z_{0}\right)^{2},$ (4a) $\mbox{or, }C_{T}^{2}=\left(\Delta
z_{0}\right)^{4/3}\Gamma^{2}.$ (4b)

It is needless to say that Eq. (4b) is virtually identical to Eq. (3) as long
as $\Delta z_{0}$ is proportional to $L_{0}$. In a latter section of this
paper, we will show that an equation similar to Eq. (4b) can also be derived
by making use of the vertical scaling characteristics of temperature and wind
speed in the buoyancy-range ($\Delta z\gg\Lambda_{0}$).

Even though the heuristic arguments can in general provide valuable insights,
in turbulence research, more rigorous results can be obtained via variance and
flux budget equations. Tatarskii Tatarskii (1971) made great strides in this
direction as discussed next.

###  II.2 Turbulence Variance Budget Equations

Tatarskii’s derivation for Eq. (3) is for a specific type of turbulent flow:
uniform shear flow with an imposed temperature gradient. He further
(implicitly) considered temperature to be a passive scalar. This assumption
implies that the dynamical evolution of temperature is driven by the velocity
field; however, temperature does not modulate the velocity field in any
manner. A direct consequence of this assumption is that the buoyancy terms are
neglected in the Navier-Stokes momentum and TKE equations.

In line with non-buoyant, shear flow turbulence literature, Tatarskii assumed
the inverse of velocity shear to be a characteristic time-scale; where,
velocity shear ($S$) is:

$S=\left(\frac{\partial\overline{u}}{\partial z}\right).$ (5)

Here, $\overline{u}$ is the velocity component along the mean wind direction.
If $L_{0}$ is the characteristic length scale, then the velocity scale is
simply: $L_{0}S$. Then, the eddy viscosity coefficient ($K_{M}$), commonly
parameterized as a product of characteristic length and velocity scales, can
be expressed as:

$K_{M}=L_{0}\left(L_{0}S\right)=L_{0}^{2}S.$ (6)

This is Eq. (14.23) of Tatarskii (1971).

Under the assumption of steady-state and horizontal homogeneity, the
simplified budget equation for TKE ($\overline{e}$) can be written as:

$\overline{\varepsilon}=-\left(\overline{u^{\prime}w^{\prime}}\right)S.$ (7)

Here, the left hand and right hand sides of this equation represent
(molecular) dissipation and shear production of turbulence, respectively. The
along-wind component of the momentum flux is
$\overline{u^{\prime}w^{\prime}}$. As mentioned earlier, owing to the passive
scalar assumption, the buoyancy term is not included in Eq. (7). Furthermore,
the terms with secondary importance (e.g., turbulent transport) are neglected.

According to the K-theory, based on the celebrated hypothesis of Boussinesq in
1877, the momentum flux can be approximated as follows:

$\overline{u^{\prime}w^{\prime}}=-K_{M}S.$ (8)

By combining Eqs. (7) and (8), we get:

$\overline{\varepsilon}=K_{M}S^{2}.$ (9)

This is Eq. (14.21) of Tatarskii (1971).

Similar to Eq. (7), the simplified budget equation of potential temperature
variance ($\sigma_{\theta}^{2}$) can be written as follows:

$\overline{N}_{\theta}=-\left(\overline{w^{\prime}\theta^{\prime}}\right)\Gamma,$
(10)

where $\overline{w^{\prime}\theta^{\prime}}$ denotes the sensible heat flux.
Once again, by using the K-theory, we can write:

$\overline{w^{\prime}\theta^{\prime}}=-K_{H}\Gamma.$ (11)

Here $K_{H}$ is the eddy-diffusivity coefficient and the ratio of $K_{M}$ and
$K_{H}$ is called the turbulent Prandtl number ($Pr_{t}$). By combining Eqs.
(10) and (11), we get:

$\overline{N}_{\theta}=K_{H}\Gamma^{2}=\left(\frac{K_{M}}{Pr_{t}}\right)\Gamma^{2}.$
(12)

This is Eq. (14.20) of Tatarskii (1971).

Now, via Eqs. (2b), (6), (9), and (12), we can deduce Eq. (3) as follows:

$C_{T}^{2}=c\left(K_{M}S^{2}\right)^{-1/3}\left(\frac{K_{M}}{Pr_{t}}\right)\Gamma^{2}=\left(\frac{c}{Pr_{t}}\right)\left(\frac{K_{M}}{S}\right)^{2/3}\Gamma^{2}=\left(\frac{c}{Pr_{t}}\right)L_{0}^{4/3}\Gamma^{2}.$
(13)

These equations are documented as Eqs. (14.22) and (14.24) in Tatarskii
(1971).

It is well accepted in turbulence literature that both shear and buoyancy
deform the larger eddies more compared to the smaller ones (e.g., Smyth and
Moum, 2000; Basu et al., 2021). Corrsin Corrsin (1958) postulated that the
eddies smaller than
$L_{C}=\left(\frac{\overline{\varepsilon}}{S^{3}}\right)^{1/2}$ are not
affected by shear. It is important to recognize that Tatarskii’s length scale
$L_{0}$ is actually Corrsin’s length scale $L_{C}$. Based on Eqs. (6) and (9),
we can show that:

$L_{0}=\left(\frac{K_{M}}{S}\right)^{1/2}=\left(\frac{\overline{\varepsilon}}{S^{3}}\right)^{1/2}=L_{C}.$
(14)

In shear flow turbulence, the eddies can be assumed to be isotropic if they
are smaller than $L_{C}$ Corrsin (1958); Basu et al. (2021). However, for
buoyancy-dominated flows, $L_{C}$ (and $L_{0}$) may not be a relevant length
scale. Tatarskii realized this possibility and advocated for a specific
buoyancy length scale in Section 17 of Tatarskii (1971). In the following
section, we discuss Tatarskii’s strategy and point out its limitations.

##  III Buoyancy-Range Scaling

###  III.1 Hypotheses of Bolgiano and Obukhov

Over the years, various scaling-based hypotheses have been proposed for the
buoyancy-range (scales larger than the inertial-range) of turbulence. One
popular hypothesis was independently formulated by Bolgiano Bolgiano Jr.
(1959, 1962) and Obukhov Obukhov (1959). They (henceforth Bolgiano-Obukhov or
BO) postulated that in the buoyancy range (i.e., $\Delta z\gg\Lambda_{0}$),
the relevant variables are $\overline{N}_{\theta}$, and $\beta$; where the
buoyancy parameter ($\beta$) is defined as follows:

$\beta=\left(\frac{g}{\Theta_{0}}\right).$ (15)

Here the gravitational acceleration is denoted by $g$ and $\Theta_{0}$ is a
reference potential temperature. Via simple dimensional analysis, BO derived
the following relationship for second-order structure function of potential
temperature in the buoyancy range:

$S_{2\theta}^{BO}\sim\left(\overline{N}_{\theta}\right)^{4/5}\left(\beta\right)^{-2/5}\left(\Delta
z\right)^{2/5}.$ (16)

A schematic of this relationship is shown in the left panel of Fig. 1. In the
right panel of this figure, the BO formulation for potential temperature
spectra is depicted.

Figure 1: Scaling regimes of atmospheric turbulence. The left and right panels
represent second-order structure function and spectra of potential
temperature, respectively. The scaling hypothesis by Kolmogorov-Obukhov-
Corrsin (KOC) holds in the inertial-range. However, various competing
hypotheses exist for the buoyancy range. The scaling relationships by
Bolgiano-Obukhov (BO) and Shur-Lumley (SL) are compared in these schematics.

From the schematics in Fig. 1, it is evident that at the cross-over point
between the inertial-range and the buoyancy-range (i.e., $\Delta
z=\Lambda_{0}^{BO}$), both Eqs. (1) and (16) should hold. As a result, we
have:

$\left(\overline{\varepsilon}\right)^{-1/3}\overline{N}_{\theta}\left(\Lambda_{0}^{BO}\right)^{2/3}\sim\left(\overline{N}_{\theta}\right)^{4/5}\left(\beta\right)^{-2/5}\left(\Lambda_{0}^{BO}\right)^{2/5}.$
(17a) By algebraic manipulation, we get a length scale:
$\Lambda_{0}^{BO}\equiv\left(\overline{\varepsilon}\right)^{5/4}\left(\overline{N}_{\theta}\right)^{-3/4}\left(\beta\right)^{-3/2}.$
(17b) In the literature, this specific length scale is known as the Bolgiano-
Obukhov length scale. Tatarskii referred to this (buoyancy) length scale in
Eq. (17.1) of Tatarskii (1971).

Tatarskii Tatarskii (1971) recognized that $L_{0}$ is essentially a free
parameter in his formulation [i.e., Eq. (3)] and should be prescribed or
estimated separately. In order to circumvent this limitation, Tatarskii
derived a formal relationship between $L_{0}$ and a more physically-based
length scale $\Lambda_{0}^{BO}$ as discussed below. Since Eq. (7) does not
include the buoyancy term, as a first step, Tatarskii used a revised budget
equation:

$\overline{\varepsilon}=-\left(\overline{u^{\prime}w^{\prime}}\right)S+\left(\overline{w^{\prime}\theta^{\prime}}\right)\beta.$
(18a) The second term on the right hand side signifies production (unstable
condition) or dissipation (stably stratified condition) of TKE due to the
buoyancy effects. Using the K-theory [refer to Eqs. (8) and (11)] and the
definition of turbulent Prandtl number ($Pr_{t}$), this equation can be
rewritten as follows:
$\overline{\varepsilon}=K_{M}\left(S^{2}-\frac{\Gamma\beta}{Pr_{t}}\right)=K_{M}S^{2}\left(1-\frac{Ri_{g}}{Pr_{t}}\right),$
(18b) where $Ri_{g}$ $\left(=\beta\Gamma/S^{2}\right)$ is the gradient
Richardson number. It is a non-dimensional variable which quantifies the
strength of atmospheric stability. Non-buoyant conditions are characterized by
$Ri_{g}=0$.

Now, by utilizing Eqs. (12), (14), (17b), and (18b), the length scale
$\Lambda_{0}^{BO}$ can be re-written as:

$\Lambda_{0}^{BO}=\left(\frac{K_{M}}{S}\right)^{1/2}\left[\frac{Pr_{t}^{3/4}\left(1-Ri_{g}/Pr_{t}\right)^{5/4}}{Ri_{g}^{3/2}}\right]=L_{0}\left[\frac{Pr_{t}^{3/4}\left(1-Ri_{g}/Pr_{t}\right)^{5/4}}{Ri_{g}^{3/2}}\right],$
(19a) or,
$L_{0}=\Lambda_{0}^{BO}\left[\frac{Ri_{g}^{3/2}}{Pr_{t}^{3/4}\left(1-Ri_{g}/Pr_{t}\right)^{5/4}}\right].$
(19b)

This is essentially Eq. (17.10) of Tatarskii (1971).

In principle, if one can measure $\Lambda_{0}^{BO}$, then $L_{0}$ can be
estimated via Eq. (19b). However, direct estimation of $\Lambda_{0}^{BO}$, in
our atmosphere, is challenging (if not impossible) since it requires reliable
estimates of both $\overline{\varepsilon}$ and $\overline{N}_{\theta}$. Apart
from this practical limitation, there are more fundamental issues.

In the literature, only a handful of laboratory and numerical studies of
idealized flows (e.g., Boffetta et al., 2012; Kumar et al., 2014; Niemela et
al., 2000; Verma, 2018) have ever documented the existence of the BO scaling.
However, to the best of our knowledge, the observational support for this
scaling hypothesis in our atmosphere is non-existent. Thus, the usage of
$\Lambda_{0}^{BO}$ as an outer (cross-over) scale is questionable.

Another critical issue is the incompatibility of $\Lambda_{0}^{BO}$ with Eq.
(3). Using Eq. (2b), it is straightforward to manipulate Eq. (17b) to arrive
at:

$C_{T}^{2}=\left(\overline{N}_{\theta}\right)^{4/5}\left(\beta\right)^{-2/5}\left(\Lambda_{0}^{BO}\right)^{-4/15}.$
(20)

It is needless to say that this equation is distinctively different from Eq.
(3). In other words, Tatarskii’s strategy to couple the BO hypothesis with his
own $C_{T}^{2}$ formulation is not justified. However, we can make use of a
different scaling formalism to resolve this inconsistency as elaborated below.

###  III.2 Hypotheses of Shur, Lumley, Monin, and Weinstock

In the past, several studies (e.g., Cot and Barat, 1989; Hovde et al., 2011;
Nastrom et al., 1997; Tsuda et al., 1991) have performed 1-D spectral and
structure function analyses of vertical (potential) temperature and wind speed
profiles measured by radiosondes and dropsondes in the atmosphere. They often
reported a quasi-universal $\kappa_{z}^{-3}$ spectral scaling behavior over
extended vertical ranges; where $\kappa_{z}$ is the vertical wavenumber.
Similarly, in the physical space, the second-order structure function was
found to scale as $\left(\Delta z\right)^{2}$. Please refer to the schematics
in Fig. 1 for a summary of various scaling regimes and associated hypotheses.

To explain the observed vertical scaling behaviors of (potential) temperature
and wind speed profiles, several competing hypotheses have been proposed in
the literature (e.g., Dewan, 1997; Lumley, 1964; Monin, 1965; Shur, 1962;
Weinstock, 1985). In this paper, we discuss the hypothesis of Shur Shur (1962)
and Lumley Lumley (1964) (henceforth the SL hypothesis) because of its
simplicity. Monin Monin (1965) further simplified the SL hypothesis by using
dimensional analysis and we follow his approach. He assumed that, in the
buoyancy-range of turbulence, $\beta$ and $\Gamma$ are the only relevant
variables. Then, via dimensional analysis, it is trivial to show that the
second-order structure functions should behave as follows:

$S_{2u}^{SL}\sim\beta\Gamma\left(\Delta z\right)^{2},$ (21a)
$S_{2\theta}^{SL}\sim\beta^{0}\Gamma^{2}\left(\Delta
z\right)^{2}=\Gamma^{2}\left(\Delta z\right)^{2}.$ (21b)

According to Kolmogorov’s hypothesis Kolmogorov (1941), the following
relationship holds in the inertial-range:

$S_{2u}^{K41}\sim\varepsilon^{2/3}\left(\Delta z\right)^{2/3}.$ (22)

By matching Eq. (21a) with Eq. (22) at the cross-over point (i.e., $\Delta
z=\Lambda_{0}^{SL}$), we get:

$\displaystyle\beta\Gamma\left(\Lambda_{0}^{SL}\right)^{2}$
$\displaystyle\equiv\varepsilon^{2/3}\left(\Lambda_{0}^{SL}\right)^{2/3},$
(23a) $\displaystyle\mbox{or, \hskip
7.22743pt}\left(\Lambda_{0}^{SL}\right)^{4/3}$
$\displaystyle\equiv\left(\frac{\varepsilon^{2/3}}{\beta\Gamma}\right)=\left(\frac{\varepsilon^{2/3}}{N_{BV}^{2}}\right),$
(23b) $\displaystyle\mbox{Therefore, \hskip 7.22743pt}\Lambda_{0}^{SL}$
$\displaystyle\equiv\left(\frac{\varepsilon}{N_{BV}^{3}}\right)^{1/2}=L_{OZ}.$
(23c)

Here $N_{BV}$ is the so-called Brunt-Väisälä frequency. The length scale
$L_{OZ}$ was first proposed by Ozmidov Ozmidov (1965).

In a similar manner, by matching Eq. (21b) against Eq. (2a) at
$\Lambda_{0}^{SL}$, one can deduce:

$C_{T}^{2}\left(\Lambda_{0}^{SL}\right)^{2/3}\equiv\Gamma^{2}\left(\Lambda_{0}^{SL}\right)^{2}.$
(24a) Thus,
$C_{T}^{2}\equiv\left(\Lambda_{0}^{SL}\right)^{4/3}\Gamma^{2}=\left(L_{OZ}\right)^{4/3}\Gamma^{2}.$
(24b)

This scaling-based equation is almost identical to Eq. (3) if $L_{0}$ is
proportional to $L_{OZ}$ and $Pr_{t}$ is on the order of one.

In summary, the Bolgiano-Obukhov scaling is not in agreement with Tatarskii’s
$C_{T}^{2}$ formulation. Instead, in a non-rigorous manner, Tatarskii’s
formulation can be deduced from the well-known vertical scaling
characteristics of (potential) temperature and wind speed profiles. In the
following section, we will take a completely different route. Based on the
rigorous turbulence variance and flux budget equations, we will derive a
revised $C_{T}^{2}$ formulation along with a physically-based length scale
formulation. We want to emphasize that in contrast to Tatarskii’s approach,
our formulation does not require an ad-hoc prescription of an outer length
scale; the length scale is an integral part of our analytical derivations.

##  IV Revised $C_{T}^{2}$ Formulation

In a recent study Basu and Holtslag (2021), we have derived closed-form
solutions for a characteristic length scale (denoted as $L_{X}$) and turbulent
Prandtl number ($Pr_{t}$) based on the conventional budget equations. For
brevity, we only report the key findings from Basu and Holtslag (2021) which
are directly relevant for the present paper. In Basu and Holtslag (2021), we
used Eqs. (10), (18b), and an additional budget equation for sensible heat
flux to deduce:

$\displaystyle
L_{X}=\left(\frac{\sqrt{Pr_{t0}Pr_{t}}}{c_{\theta}}\right)\left(\frac{\sigma_{\theta}}{\Gamma}\right),$
(25)

where $L_{X}$ is the characteristic length scale for stably stratified flows.
The standard deviation of potential temperature is $\sigma_{\theta}$. The
turbulent Prandtl number for non-buoyant flows is denoted by $Pr_{t0}$; it is
typically assumed to be equal to 0.85. The coefficient $c_{\theta}$ is
approximately equal to 2.

In Basu and Holtslag (2021), the dissipation rates of TKE ($\overline{e}$) and
variance of temperature ($\sigma_{\theta}^{2}$) are derived to be respectively
as follows:

$\overline{\varepsilon}=\left(\frac{\sigma_{w}^{3}}{c_{w}^{3}L_{X}}\right),$
(26a) and
$\overline{\chi}_{\theta}=\left(\frac{2Pr_{t0}}{c_{w}c_{\theta}^{2}}\right)\left(\frac{\sigma_{w}\sigma_{\theta}^{2}}{L_{X}}\right).$
(26b)

Here the standard deviation of vertical velocity is $\sigma_{w}$. The
coefficient $c_{w}$ is approximately equal to 1.25. From Eqs. (2b), (26a), and
(26b), we get:

$C_{T}^{2}=\left(\frac{c}{2}\right)\left(\frac{c_{w}L_{X}^{1/3}}{\sigma_{w}}\right)\left(\frac{2Pr_{t0}}{c_{w}c_{\theta}^{2}}\right)\left(\frac{\sigma_{w}\sigma_{\theta}^{2}}{L_{X}}\right)=\left(\frac{cPr_{t0}}{c_{\theta}^{2}}\right)\left(\frac{\sigma_{\theta}^{2}}{L_{X}^{2/3}}\right).$
(27)

By substituting $\sigma_{\theta}$ from Eq. (25) to Eq. (27), we find:

$C_{T}^{2}=\left(\frac{cPr_{t0}}{c_{\theta}^{2}}\right)L_{X}^{4/3}\Gamma^{2}=\left(\frac{c}{Pr_{t}}\right)L_{X}^{4/3}\Gamma^{2}.$
(28)

This equation is identical to Eq. (3) with one important difference. Here the
length scale $L_{X}$ is defined by Eq. (25), and is not a free parameter.
Several well-known length scales are explicitly related to $L_{X}$; please
refer to Appendix 1 and Basu and Holtslag (2021) for further details.

Earlier, we mentioned that in the atmospheric optics literature the value of
$Pr_{t}$ is frequently assumed to be equal to unity. However, there is ample
evidence in the fluid dynamics and atmospheric turbulence literature that
$Pr_{t}$ monotonically increases with increasing values of $Ri_{g}$. In this
context, a physically-based formulation was derived by Basu and Holtslag
(2021).

##  V $C_{T}^{2}$ in Convective Mixed Layer

Convective (unstable) condition typically occur over land during daytime
conditions. At offshore locations, such conditions prevail when air
temperature is colder than the underlying sea-surface temperature. Under
convective conditions, due to intense turbulent diffusion, the so-called
’mixed’ layer (ML) develops a few tens of meters above the land or sea-
surface. Due to turbulent mixing, (potential) temperature, moisture, and other
meteorological variables become uniformly distributed in the ML. In other
words, in the ML, $\Gamma=\frac{\partial\overline{\theta}}{\partial z}\approx
0$. As a result, Tatarskii’s equation [i.e., Eq. (3)] and our proposed
formulation [i.e., Eq. (28)] incorrectly predict $C_{T}^{2}\approx 0$ in the
ML. The reasons behind this unphysical prediction are discussed below.

In the ML, large coherent structures (called thermals, plumes) are primarily
responsible for turbulent mixing. These structures originate near the surface
and can reach the top of the ML (on the order of a km or so) in a matter of 10
to 15 minutes. During their ascent, these structures cause ‘non-local’ mixing.
As a result, turbulent fluxes are no longer proportional to the local mean
gradients. In other words, the application of K-theory is not tenable in the
ML. Hence, Eqs. (8) and (11) are not valid in the ML Holtslag and Moeng
(1991). Furthermore, the simplified variance and flux budget equations [e.g.,
Eq. (18b)] are not suitable for the ML. Additional non-local transport terms
should be included in them. Unfortunately, such modifications make these
equations analytically intractable.

We recommend the readers to use Eq. (3) and/or Eq. (28) only for stably
stratified conditions. Such conditions are omni-present in the free atmosphere
(i.e., the layer above the atmospheric boundary layer). Also, nocturnal
boundary layers over land are commonly stably stratified. For convective ML, a
similarity-based formulation by Kaimal et al. Kaimal et al. (1976) or its
variants could be used. On this topic, an interesting study was recently
published by Luce et al. (2020). However, more research is highly desired in
this arena.

##  VI Concluding Remarks

Fifty years ago, Tatarskii developed a simple $C_{T}^{2}$ formulation which
has found wide usage in a range of scientific and engineering disciplines,
from astronomy to free-space optical communication. In this paper, we revisit
this formulation and point out its limitations. We then propose a revised
$C_{T}^{2}$ formulation based on turbulence variance and flux budget
equations. In contrast to Tatarskii’s equation, our formulation includes a
novel length scale which is physically-based. Since this length scale is
simply dependent on certain variances and mean gradients, it can be estimated
from observational data. Such applications will be reported in a separate
publication.

###### Acknowledgements.

We are truly grateful to Hubert Luce and Lakshmi Kantha for thought-provoking
communications and for providing constructive feedback on our work.

## Appendix 1: Inter-relationships of Length Scales

In addition to Eq. (25), two alternative definitions of $L_{X}$ were proposed
by Basu and Holtslag (2021):

$\displaystyle L_{X}$
$\displaystyle=c_{H}\left(\frac{\sigma_{w}}{S}\right)\left(\frac{1}{\sqrt{1/G-Ri_{g}/Pr_{t}}}\right),$
(29a)
$\displaystyle=c_{H}\left(\frac{\sigma_{w}}{N_{BV}}\right)\left(\frac{\sqrt{Ri_{g}}}{\sqrt{1/G-Ri_{g}/Pr_{t}}}\right).$
(29b)

Where the coefficient $c_{H}$ is approximately equal to 0.8. The term $G$,
called a ‘growth factor’ Schumann and Gerz (1995); Basu and Holtslag (2021),
represents the ratio of the production and dissipation terms of the TKE
equation. For weakly/moderately stable condition, $G$ equals to one. Under
strongly stratified condition $G$ is expected to be smaller than one. Basu and
Holtslag Basu and Holtslag (2021) proposed the following heuristic
parameterization for $G$:

$G=\min\left(1,Ri_{g}^{-1}\right).$ (30)

An unique relationship between $L_{X}$ and the Corrsin’s length scale
($L_{C}$) can be established by utilizing Eq. (14), Eq. (26a) and Eq. (29a):

$\displaystyle\frac{L_{X}}{L_{C}}$
$\displaystyle=\left(\frac{1}{\sqrt{1/G-Ri_{g}/Pr_{t}}}\right)^{3/2}.$ (31)

When $Ri_{g}\to 0$, $G=1$ and $L_{X}\to L_{C}$.

A different relationship can be deduced between the Ozmidov length scale
($L_{OZ}$) and $L_{X}$ via Eq. (23c), Eq. (26a) and Eq. (29b):

$\displaystyle\frac{L_{X}}{L_{OZ}}$
$\displaystyle=\left(\frac{\sqrt{Ri_{g}}}{\sqrt{1/G-Ri_{g}/Pr_{t}}}\right)^{3/2}.$
(32)

For $Ri_{g}\gg 1$, if $G$ is proportional to $Ri_{g}^{-1}$, then $L_{X}\to
L_{OZ}$.

In summary, the length scale $L_{X}$ acts as a smooth interpolator between two
limiting length scales: $L_{C}$ and $L_{OZ}$. For this reason, it has been
called a hybrid length scale.

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# Co-Design of Assistive Robotics with Additive Manufacturing and Cyber-
Physical Modularity to Improve Trust

Alexandre Colle1, Ronnie Smith1, Scott MacLeod2 and Mauro Dragone2 This work
was supported in part by the Engineering and Physical Sciences Research
Council (EPSRC) Centre for Doctoral Training (CDT) in Robotics and Autonomous
Systems (grants EP/L016834/1 and EP/S023208/1) at Heriot-Watt University and
The University of Edinburgh.1Alexandre Colle and Ronnie Smith are with the
Edinburgh Centre for Robotics at Heriot-Watt University and The University of
Edinburgh, Edinburgh UK. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>MacLeod and Mauro Dragone are with Heriot-Wartt
University and affiliated with the Edinburgh Centre for Robotics, Edinburgh,
UK. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Robotics and automation have the potential to significantly improve quality of
life for people with assistive needs and their carers. Adoption of such
technologies at this point in time is far from widespread. This paper presents
a novel approach to the design of highly customisable robotic concepts,
embracing modularity and a co-design process to increase the involvement of
end-users in the development life cycle. We discuss this process within the
context of an elderly care use case. Using design methodology and additive
manufacturing, we outline how key stakeholders can be involved from initial
conception through to integration of the final product within their
environments. In future work, we will apply this process to demonstrate the
effectiveness of our approach for improving long-term acceptance and trust of
robotic technology in care contexts.

##  I Introduction

Health and elderly care systems in many developed countries have been facing
significant pressure for several years [1]: budget cuts, a lack of human
resources, and a growing elderly demographic have tested the boundaries of
existing care systems [2]. The recent COVID-19 crisis has increased this
pressure further, putting the lives of both care workers and care recipients,
at risk.

We have seen robotics applied to various aspects of care, such as in
minimally-invasive surgery and in automated cleaning of facilities [3]. The
integration of robotics into existing care practice nonetheless remains
difficult for several reasons, which we explore herein.

Providing high-quality care, whether for healthcare in general or for elderly
care in particular, requires significant human effort. The demands and
requirements of care staff may differ greatly from those of patients and
residents [2]. Furthermore, the introduction of new technology can be
difficult to accept, which can lead to misuse of or outright refusal to
interact with Assistive Technology (AT).

Current robotic platforms, in the context of care, tend to provide very
specific solutions to users. These robots represent completed products, with
development teams often removed from the reality of users. These devices are
subject to manufacturing processes and commercialisation constraints, often in
conflict with systemic problems of stakeholders in the context of care [4].
The initial manufacturing costs are often substantial.

Understanding the needs and wants of stakeholders is fundamental to the
development of meaningful AT. User-Centred Design (UCD) and ‘Co-Design’ can
improve the design of assistive technology for elderly care, by placing the
users at the core of the design process. At the same time, co-design processes
encourage human interactions and social contact, thus increasing trust between
researchers and participants [5][6]. However, researchers need to adapt to the
specific requirements of older users with specific tools appropriate for an
audience less well-versed in technology. Co-design has been adopted largely in
recent years in the field of Human-Robot Interaction (HRI) [7][8][9][10],
however, few projects have led to the design of tangible robots with a
practical use case. Service robots remain very complex technological systems.
Their development requires multidisciplinary teams, working closely to achieve
a common goal with minimum resources.

Research in Artificial Intelligence (AI) has led to improved cognitive
capacity in robots. In contrast, as their minds evolve, their physical form
remains largely unaltered [11]. The current way robots are designed is
prohibitive of physical alteration and discourages active personalisation.
Customisation, the process of letting users actively modify the robot
appearance, includes users further in the development of the robot in the
long-term. When users are involved with customisation, it increases customer
satisfaction, integration, and loyalty [12]. For robotics, this process can
have clear benefits for social interactions [13], and encourage active
participation and use of a co-created, customisable, device.

According to Gibson et al. [14], in the context of dementia care, users tends
to “tame” technological artefacts by adapting them to their individual needs.
Gibson et al. refer to this practice as bricolage: where carers use creative
and non-conventional ways to use AT to adapt to local needs. They also
highlight the frustration of these same carers, where their specific input
does not materialise in final products. As such, bricolage highlights the
importance of constant design iterations in the context of care, in order to
create devices matching the vast array of needs from both carers and those
receiving care. It also emphasises the risk of developing narrow technological
solutions in conflict with the reality of users requirements. In contrast, the
concept of bricolage encourages provider of AT solutions to facilitate user-
driven customisation, instead of current standardised solutions [14]. Above
all, it validates the need for a systemic method such as co-design, to capture
this iterative process, and translate these suggestions into tangible
outcomes.

We hypothesise that the outcomes of co-design can be improved through the use
of Additive Manufacturing (AM) and a modular approach to robot development,
ultimately improving acceptance and trust in the robot.

Our long-term vision is to develop a framework to guide the development of a
robot combining co-design and AM. This paper explores the role of co-design
and customisation as a mediator to improve the long-term acceptance of robots
in a care setting. We also discuss the role of AM in improving the conception
of modular robots to facilitate the participatory design work.

Research Question: how can we more effectively design these new types of
robots?

##  II Benefits of Modular Robots

Current commercial robots have been effectively adopted in various
environments. The acquisition of these machines represents a major investment
due to their complexity, however, their flexibility at executing tasks has
great advantages. A new generation of service robots is challenging the
fundamental nature of these machines. Contrary to traditional robots, modular
robots offer a better solution to unpredictable and evolving tasks [15]. For
example, a stackable, re-configurable and adaptable concept for monitoring
oil-rigs named Limpet II [16], allows users to swap modules to execute
different tasks. Instead of a uniform and expensive solution provided by
traditional robots, the Limpet II creates a low cost and adaptable solution to
achieve multiple tasks within a complex environment [16]. Modular robots raise
the prospect of fully adaptable devices, in constant evolution with users
needs.

###  II-A Improving Flexibility

The development process behind individual robots implies strong financial
capabilities from robotics companies [17]. Research and Development (R&D)
represents a substantial cost, and market readiness absorbs a large part of
the initial resources. Additionally, a robotics company depends on the
capacity and willingness of a manufacturing partner to produce final products
within a specific price range. Thus, Social robots, in the context of
healthcare, suffer from a lack of flexibility in product development because
the manufacturing process depends on traditional manufacturing methods.

Progress in AM techniques and the improvement of Computer-Aided Design (CAD)
software capabilities are transforming manufacturing. AM, is “the processes
that add some materials to the previous surface via different deposition
techniques that lead to different part quality, density, and geometrical
accuracy” [18]. This ‘Industry 4.0’ [18] is faster, more sustainable [19], and
more efficient [20]. CAD allows for greater team collaboration regardless of
individual locations, while managing and optimising extremely complex designs
[21]. In conjunction with CAD, AM is breaking the chain of traditional
manufacturing. As a result, three-dimensional (3D) printing supports rapid
prototyping and local manufacturing of those designs [22].

AM and CAD create new channels to develop robots with flexibility as a
principle of development. As AM technology improves, the development of
flexible robotics can reap the benefits by applying the advancements to areas
such as rapid prototyping and manufacturing. More generally, modular robotics
is getting more attention from research and the private sector for their
advantages in terms of “reconfigurability, reusability, and ease in
manufacturing” [23][16]. Modularity can take many shapes and forms and depends
on the specific area of use of the robot. Ultimately, a clear understanding of
the context of use will inform how modularity should be applied.

###  II-B Improving Acceptance

Maintaining acceptance levels of any given robot in the long-term,
particularly in a public environment, remains a challenge for HRI. One of the
goals of HRI is to improve the long-term acceptance of social robots
[13][24][25][26][27]. Robots as an object are perceived by different
individuals in different ways. From patients to health workers, robots can
either be seen as a threat or an opportunity. The way robots are developed at
the moment does not encourage active public participation in the eventual
behaviours and overall appearance of the robot.

Of course, robots are not positively perceived by everyone. Individual taste
varies from one person to another, as aesthetic judgement is made out of both
conscious and subconscious decisions [28]. Accordingly, the opinion of an
audience will vary greatly when confronted with robots with a uniform design.
Moreover, a moving agent will increase anthropomorphic interpretation and the
impression of agency [29]. Depending on the context, these interactions can
have overwhelmingly positive or negative outcomes such as fear and total
rejection [30].

Technology acceptance differs between age groups and the integration of
complex technology varies with the context of use. For an end-user, taking an
active part in the design and functionality of a complex device like a robot
improves the chances of acceptance and consistent long-term use [31][32].
Moreover, giving users the ability to engage in personalisation encourages
them to adapt robots to their specific circumstances [31], improving
relationships with the device at the same time. Peine et al. [31] insist
technology must be domesticated or “tamed” and that “Users are more likely to
become co-designers of technology to fit complex technical arrangements into
their lives”. The motivations behind ‘bricolage’ and personalisation need to
be captured, analysed and transcribed into concrete outcomes, for both the
benefit of users and developers of the robots.

Current robotic platforms are not designed to let users actively personalise
their appearance and function. Modularity must be considered at the conception
of the device, its implication in terms of hardware and software are very
complex [23] and cannot be treated as a cosmetic add-on.

###  II-C Co-Design

The process of design thinking, as a support tool, allows for a holistic
approach to problem-solving and product development [33][34][35]. co-design,
naturally involves stakeholders and [36][37] allows for great participation of
the users involves.

Co-design is executed in several stages, consisting of generally these main
steps [38]:

  1. 1.

Empathise with users;

  2. 2.

Define problems;

  3. 3.

Ideate (idea generation);

  4. 4.

Prototype concept;

  5. 5.

Test and iterate.

Applied in the context of robotics, Compagna et al. used a participatory
approach for the development of a service robot in [39] elderly care. Their
three years study use the stages of co-design to inform the development of
their robot. The results ultimately recommended a substantial increase in
usability trials combined with better use of rapid prototyping. Additionally,
they recommended to encourage direct relations between developers and
stakeholders and avoid intermediary actors in the process.

Initially, the thought of robots may disturb the public ability to imagine
rationally what could be an autonomous device[40][41] in their environments.
The influence of science fiction raises people’s expectations of the real
capability of machines. Moreover, many people experience feelings of fear and
aversion to technology when confronted with the thoughts of robots[42]. The
label ‘robot’ can be misinterpreted, therefore there are some cases where a
different denomination [43] will ease people’s imagination.

Additionally, broader concerns can create additional frictions. For
stakeholders in the context of ageing, being seen as a sick patient in need of
care will raise suspicions and have a negative impact on a co-design
initiative [42]. On the contrary, if they become active participants working
for their own benefits and potentially helping others, they will then
appropriate the project for themselves and become far deeper involved as
participants [31]. The elderly want to be treated like normal citizens, and
considered fairly, with their own needs [44][31]. By contrast, carers will
have personal concerns against automation. They worry about the future of
their career [45] and towards the people they care about. They believe more
automation will lead to more unemployment and patients being neglected.
Stakeholders across the board must be understood and we must adapt our
approach to anticipate negative feelings. The goal of this project is to allow
them to create their tools to improve their work and well-being and at the
same time improve trust in robots.

##  III A Modular Robotics Platform for Co-Design

Having reviewed the benefits and best-practice of co-design, we now lay out
our concept of pairing it with a modular robotic platform and additive
manufacturing to enable rapid development and manufacture of co-designed
modules. We posit that this will enable significantly tighter iterative cycles
of feedback and development, therefore enabling sustained user engagement in
the process as the products of their feedback is more quickly realised (see
Figure 1).

Figure 1: Integration of co-design methods and Additive Manufacturing
development strategy.

We envisage that such a platform to enhance this process would consist of a
‘base’ robot with minimal functionality, but to which modules that provide
additional functionality can be readily attached. There is therefore a fine
balance to be struck between the functionality provided by the base and the
range of functionality that can be provided through modules.

In this scenario, the platform will become a combination of top-down
(prescribed) and bottom-up (learned, through co-design) design [46]. The
balance between the two types of design is of the utmost importance, but there
should be a bias towards the bottom-up approach. As such, a requirement for
this modular platform is that the base should provide minimal functionality,
the essential traits to be considered a ‘robot’, and can incorporate
additional hardware that might be useful (e.g. computational resources,
cameras, speakers) for use by modules, without being prescriptive about what
that use is.

In terms of hardware and software, this means the platform must be flexible
such that a newly developed module can be easily attached the to the base and
the behaviour and functionality of the module be realised in software. This
should draw from concepts in Component-Based Software Engineering (CBSE) in
that new features can be gained through ‘plug and play’ of new modules.

In contrast with traditional platforms, our method can produce, in a limited
time-frame, a realistic and deployable robotic system within users’
environments. The modular nature of the device is naturally designed for
continuous personalisation.

We are suggesting to emphasis the following steps for the development of this
device:

  * •

Method: Methods such as design thinking help structuring a project around the
same goal. It also represents a similar language when multidisciplinary teams
collaborate, by providing a central canvas for stakeholders and developers can
engage in a “design Conversation” [47];

  * •

Conception: Advancements in additive manufacturing technologies improves not
only rapid prototyping as part of the co-design method, it also allows for the
rapid manufacturing of a finished product [48]. Our goal is to use the
flexibility offered by additive manufacturing to rapidly execute the changes
highlighted during the co-design sessions. The results can offer a tangible
result and create a concrete outcome for participants. Ideas become real,
therefore increasing trust between organisation and participants [49];

  * •

Design: The appearance and behaviours of a social robot have a major impact on
how negatively or positively a robot will be perceived by users [50].
Therefore, the design of the appearance of the robot is a crucial part of the
design process. By contrast, this robot represents a new design challenge and
transgresses the current paradigm of design. As a result, we advocate a new
design principle, where we purposefully limit the final appearance of the
robot. Following our hypothesis, final users will then be encouraged to finish
the overall design of the robot by engaging in active personalisation. For
this purpose, we employ an open-ended design methodology, leaving the final
outcome forever open, never completely finished [51]. As mentioned earlier,
the shape of robots needs to evolve in relation to their cognitive capacity,
and also with the needs of the users. Both have unlimited room to grow.

##  IV Conclusion & Future Work

In this paper, we have highlighted the pitfalls of current assistive robotic
solutions, particularly for use with the elderly. We have also highlighted the
potential benefits of flexible manufacturing techniques, such as Additive
Manufacturing, and emphasised the promise of the co-design process.

On that basis, we have made recommendations about how the outcomes of co-
design of assistive robotics can be improved through iterative cycles of idea
generation, rapid development, and feedback. Furthermore, we have highlighted
how this may improve overall acceptance and trust in robotics.

In our future work, we will put our suggested process into practice as we
develop a modular robotic platform that will serve as a sandbox for the co-
design of useful assistive modules with our partners in the health and social
care industries.

The co-design process will begin with some fieldwork and observation on-site.
Co-design implies collaboration and a long-term engagement from the
researchers and the stakeholders. As such, we plan to start the process by
using semi-structured interviews to engages with health and social care
stakeholders, involving carers and care and support service users, in a direct
and open conversation. This process personifies the research, offers the
opportunity of social bond and facilitates long-term collaboration.

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