Retrospective Analysis of Micrometeorological Observations Above an Australian Wheat Crop

Abstract

We apply well-established flux–gradient relationships to deduce the aerodynamic and radiative properties of a winter wheat crop, using a neglected 1971 dataset (hourly averages), only recently resurrected as part of a historical review of precision CO₂ measurements in Australia (Pearman et al. in Hist Rec Aust Sci 28:111–125, 2017). The aerodynamic roughness length (seasonal variation between 0.07 and 0.14 of the mean crop height) and broadband albedo (seasonal variation between 0.13 and 0.23) are consistent with values published in the literature over the past 50 years. Net radiation at night is found to agree with the net longwave flux only when the dry-bulb temperature exceeds 10 °C, probably the result of dewfall on one or both of the two instruments. During the day, the sum of the four individual radiative flux components (upwards and downwards shortwave and longwave)—the composite net radiation—exceeds the directly measured net radiation, from near zero at sunrise to approximately 100 W m⁻² at maximum net radiation ≈ 600 W m⁻², viz. an underestimate in the directly measured net radiation of close to 15%. Again, this is in line with instrument comparisons made in the USA and Europe 15–25 years ago. A novel approach is used in the analysis of terms in the surface energy budget, viz., normalization of all terms by the downwelling shortwave flux. Normalization reveals, (1) near-normal frequency distributions of both the total turbulent heat flux (sensible plus latent) and the implied total storage (the residual); (2) significant diurnal variations in the total turbulent heat flux, whose standard deviations of individual values about any hourly mean during daytime are reduced significantly on those for either the sensible or latent heat flux; (3) an implied storage term with a well-defined diurnal variation, but with an overall mean value of 1% of the shortwave input. Overall, with the above results in mind, the computed momentum and heat fluxes (and also the CO₂ flux) during the daytime, at small to moderate gradient Richardson numbers, provide support for the profile approach when eddy-correlation fluxes are unavailable. Even so, possible errors due to, (1) uncertainties in the zero-plane displacement, and (2) influences of the roughness sublayer, must be borne in mind.

Appendices

Appendix 1

We take the following flux–gradient relations valid for steady-state and horizontally homogeneous conditions (e.g., Garratt 1992; Chap. 3),

$$ - \overline{w'u' } = u_{*}^{2} = K_{m} \partial u/\partial z, $$

(13)

$$ \overline{w'\theta '} = H/\rho_{a} c_{p} = {-}K_{h} \partial \varTheta /\partial z, $$

(14)

$$ \overline{w'q' } = \lambda E/\lambda \rho_{a} = {-}K_{h} \partial q/\partial z, $$

(15)

$$ \overline{w'c' } = F_{c} /\rho_{a} = {-}K_{h} \partial c/\partial z, $$

(16)

where K_m and K_h are eddy transfer coefficients for momentum, and for the scalars Ѳ (heat), q (water vapour) and c (the ${\text{CO}}_{2}$ concentration or mass mixing ratio). In the above, u_∗ is the friction velocity, H is the sensible heat flux, E is the evaporation rate ($\lambda E$ being the latent heat flux), and F_c is the vertical ${\text{CO}}_{2}$ flux. Non-dimensional forms of the vertical gradients in the inertial sublayer, and based on Monin–Obukhov similarity theory, are consistent with taking,

$$ K_{m} = ku_{ * } z/\varphi_{m } \left( {z/L} \right) $$

(17)

and

$$ K_{h} = ku_{ * } z/\varphi_{h } \left( {z/L} \right), $$

(18)

where k is the von Kármán constant, $\varphi_{c } {\text{and}} \varphi_{m}$ are stability functions, and L is the Obukhov length. Combining the above gives

$$ (kz/u_{ * } )\partial u/\partial z \, = \varphi_{m } $$

(19)

whence,

$$ H/\rho_{a} c_{p} = {-}[ku_{ * } z/\varphi_{h } ]\partial \varTheta /\partial z, $$

(20)

$$ \lambda E/\lambda \rho_{a} = {-}[ku_{ * } z/\varphi_{h } ]\partial q/\partial z. $$

(21)

$$ F_{c} /\rho = {-}[ku_{ * } z/\varphi_{h } ]\partial c/\partial z. $$

(22)

From these three last equations, we can write

$$ u_{ * } = \, (kz/\varphi_{m } )\partial u/\partial z, $$

(23)

$$ H/\rho_{a} c_{p} = - [(kz)^{2} /\varphi_{m } \varphi_{h } ]\partial u/\partial z \, \partial \varTheta /\partial z, $$

(24)

$$ \lambda E/\lambda \rho_{a} = - [(kz)^{2} /\varphi_{m } \varphi_{h } ]\partial u/\partial z \, \partial q/\partial z, $$

(25)

$$ F_{c} /\rho = \, {-}[(kz)^{2} /\varphi_{m } \varphi_{h } ]\partial u/\partial z \, \partial c/\partial z. $$

(26)

Note that the integral form of Eq. 19 or 23 produces the logarithmic wind profile in neutral conditions, with Eq. 5 in the main text being the general integral form. Equations 23–26 form the basis of the micrometeorological method common in the 1970s before eddy correlation became widely available. Then, and in the present article, we use the finite difference form since vertical differences of wind speed, dry-bulb, and wet-bulb temperatures between 1 and 2 m above the crop are the relevant measured quantities. Hence, the fluxes are based on

$$ u_{*}^{2} = \{\left[ {k\bar{z}/\Delta z} \right]^{2} /[\varphi_{m } \varphi_{m} ]\} \Delta u\Delta u, $$

(27)

$$ H = - \rho_{a} c_{p} \{\left[ {k\bar{z}/\Delta z} \right]^{2} /[\varphi_{m } \varphi_{h}]\} \Delta u\Delta \theta , $$

(28)

$$ \lambda E = - \lambda \rho_{a} \{\left[ {k\bar{z}/\Delta z} \right]^{2} /[\varphi_{m } \varphi_{h} ]\} \Delta u\Delta q, $$

(29)

$$ F_{c} = \, {-}\rho_{a} \{\left[ {k\bar{z}/\Delta z} \right]^{2} /[\varphi_{m } \varphi_{h} ]\} \Delta u\Delta c, $$

(30)

for a height increment $\Delta z$, mean height $\bar{z},$ and for which Eqs. 1–3 in the main text are the abbreviated forms. The mean height $\bar{z}$ is taken as the geometric mean height between the two selected levels, viz. $\bar{z} = \left( {z_{1} z_{2} } \right)^{0.5}$. The $\varphi_{h } {\text{and}} \varphi_{m}$ functions can be expressed in terms of the gradient virtual Richardson number Ri (e.g. see Garratt 1992, Chapter 3),

$$ Ri = \, 0.033[\partial \theta_{v } /\partial z][\partial u/\partial z]^{ - 2} $$

(31)

where the numerical factor 0.033 is simply g/θ, g is the acceleration due to gravity, and θ is the mean virtual potential temperature. With Ri evaluated using $\Delta u, \Delta \theta$, and $\Delta q,$ these well-known empirical stability functions can be readily calculated.

Thus, using the well-known relations for Ri < 0, z/L = Ri and for Ri > 0, z/L = Ri/(1 − 5 Ri), then

$$ \varphi_{h } \left( {z/L} \right) \, = \varphi_{m } (z/L)^{2} = \, \left( {1 - 16Ri} \right)^{ - 0.5} $$

(32a)

for, say, −0.5 < Ri < 0, and

$$ \varphi_{h } \left( {z/L} \right) = \varphi_{m} \left( {z/L} \right) = \left( {1 - 5Ri} \right)^{ - 1} $$

(32b)

for, say, Ri < 0.1. One arm of modern thinking (e.g., B.B. Hicks, priv. comm. 2020) argues that Monin–Obukhov similarity theory, through the flux–gradient relations given above and used herein, is only reliable (and valid) in near-neutral conditions, defined by a rather small range of Ri. The argument goes that the turbulent transfer process is modified swiftly by convective eddies as Ri becomes more negative, and is affected significantly by intermittency as Ri becomes more positive, situations for which the flux–gradient relation is a gross simplification. A second arm of modern thinking accepts the deficiencies in the flux–gradient representation of turbulent transfer, and offers this as the basis of a rigorous description of turbulent flow and its transfer properties in the roughness and inertial sublayers above an aerodynamically rough canopy (e.g., Harman and Finnigan 2007, 2008; Theeuwes et al. 2019). Even so, the theory and its application is often limited to neutral and stratified flows with, typically, − 1 < Ri < 0.1. Indeed the widespread use of Monin–Obukhov similarity theory (and hence implicitly a flux–gradient assumption, at least in part) in numerical models of the atmosphere, including numerical weather prediction, climate modelling, and large-eddy simulation, is to be noted. With few other practical options available, we have limited the use of Monin–Obukhov similarity theory in our analysis to a relatively narrow range in Ri around near-neutral conditions.

Appendix 2

We give below measured and derived quantities for most daytime hours on a sample day: day 139 (26 September 1971); crop height = 0.7 m. All measurements at hourly intervals (mostly 1-h averages) are available in Paltridge et al. (1972), which can be accessed online. For easy conversion below note that, at 15 °C, an evaporation rate of 1 mm d⁻¹ is equivalent to a latent heat flux of 28.6 W m⁻².

	0800	0900	1000	1100	1200	1300	1400	1500	1600
MEASURED
Wind direction	320	300	310	310	300	270	270	270	270
$T_{a}$ (°C)	8.0	9.4	10.6	11.1	12.5	13.1	13.7	14.0	14.3
ΔT (K)	− 0.06	− 0.17	− 0.19	− 0.09	− 0.16	− 0.06	0.08	0.16	0.22
$\Delta T_{w}$ (K)	− 0.17	− 0.28	− 0.34	− 0.29	− 0.38	− 0.35	− 0.27	− 0.24	− 0.17
$u_{1}$ (m s−1)	2.28	3.48	3.51	3.37	3.98	4.03	5.01	4.01	4.68
$u_{2}$ (m s−1)	2.72	4.15	4.16	4.01	4.72	4.79	5.96	4.80	5.60
ΔC (ppm)	1.1	1.5	1.9	2.2	2.0	1.9	1.6	1.5	1.0
$E_{lys}$ (mm ${\text{d}}^{ - 1}$)	3.1	6.1	7.9	7.0	11.3	14.6	13.4	10.4	10.4
$S_{soil}$ (W m−2)	− 8	− 1	5	6	11	11	8	5	2
$R_{n}$ (W m−2)	78	240	288	227	377	338	280	212	141
$S_{ \downarrow }$ (W m−2)	259	445	521	395	653	585	486	380	289
$S_{ \uparrow }$ (W m−2)	72	116	127	98	152	140	119	94	71
ΔL (W m−2)	− 59	− 35	− 33	− 25	− 43	− 43	− 35	− 41	− 43
DERIVED
Δq $times 10^{3}$	− 0.13	− 0.19	− 0.24	− 0.24	− 0.32	− 0.34	− 0.32	− 0.32	− 0.27
Ri	− 0.013	− 0.014	− 0.018	− 0.010	− 0.012	− 0.006	0.001	0.006	0.007
u∗ (m s−1)			0.45	0.44	0.50	0.50	0.61	0.49	0.58
H (W ${\text{m}}^{ - 2}$)	15	66	75	30	66	21	− 45	− 66	− 103
$\lambda E$ (W ${\text{m}}^{ - 2}$)	86	194	243	229	350	361	393	308	301
$F_{c}$ x $10^{7}$ ***	− 5	− 10	− 11	− 12	− 13	− 12	− 12	− 8	− 7

1. ***Note that F_c is in units of kg ${\text{CO}}_{2} \,{\text{m}}^{ - 2} {\text{s}}^{ - 1}$