Interaction of Submeso Motions in the Antarctic Stable Boundary Layer

Abstract

Submeso motions add complexities to the structure of the stable boundary layer. Such motions include horizontal meandering and gravity waves, in particular when the large-scale flow is weak. The coexistence and interaction of such submeso motions is investigated through the analysis of data collected in Antarctica, in persistent conditions of strong atmospheric stratification. Detected horizontal meandering is frequently associated with temperature oscillations characterized by similar time scales (30 min) at all levels (2, 4.5 and 10 m). In contrast, dirty gravity waves superimposed on horizontal meandering are detected only at the highest level, characterized by time scales of a few minutes. The meandering produces an energy peak in the low-frequency spectral range, well fitted by a spectral model previously proposed for low wind speeds. The coexistence of horizontal and vertical oscillations is observed in the presence of large wind-direction shifts superimposed on the gradual flow meandering. Such shifts are often related to the variation of the mean flow dynamics, but also to intermittent events, localized in time, which do not produce a variation in the mean wind direction and that are associated with sharp decreases in wind speed and temperature. The noisy gravity waves coexisting with horizontal meandering persist only for a few cycles and produce bursts of turbulent mixing close to the ground, affecting the exchange processes between the surface and the stable boundary layer. The results confirm the importance of sharp wind-direction changes at low wind speed in the stable atmosphere and suggest a possible correlation between observed gravity waves and dynamical instabilities modulated by horizontal meandering.

Appendices

Appendix 1: Spectral Model for Meandering Motions in Stable and Low-Wind-Speed Conditions

To evaluate the theoretical form of the spectrum in (3), the derivation of Mortarini and Anfossi (2015) has been followed. The normalized Eulerian spectrum, $F_{LW} \left( n \right)$, in which t is the time lag and n is the frequency, is (Pasquill 1974; Kaimal and Finnigan 1994)

$$ F_{LW} \left( n \right) = \frac{S\left( n \right)}{{\sigma^{2} }} = 4\int_{0}^{\infty } {R\left( t \right)\cos \left( {2\,\uppi\,nt} \right)dt} $$

(4)

Hence, the function

$$ F_{LW} \left( n \right) = 2p\left[ {1/\left( {p^{2} + \left( {q + 2\,\uppi\,n} \right)^{2} } \right) + 1/\left( {p^{2} + \left( {q - 2\,\uppi\,n} \right)^{2} } \right)} \right] $$

(5)

is the spectrum that corresponds to an oscillating autocorrelation function of the form

$$ R\left( t \right) = { \exp }\left( { - pt} \right){ \cos }\left( {qt} \right), $$

(6)

where $p$ and $q$ are related to the turbulence decorrelation time scale and to the meandering time scale, respectively. In non-dimensional form, $nF_{LW} \left( n \right)$, Eq. 5 presents a distinct maximum

$$ \left[ {nF_{LW} \left( n \right)} \right]_{max} = \left( {p^{2} + q^{2} } \right)^{{\frac{1}{2}}} / \left( {\uppi\,p} \right) $$

(7)

at the frequency

$$ n_{max} = \left( {p^{2} + q^{2} } \right)^{{\frac{1}{2}}} / \left( {2\uppi} \right). $$

(8)

It is easily seen that the slope of the asymptotic behaviour in Eq. 5 is − 2 and not the prescribed − 5/3. Equation 5 correctly describes the behaviour of the low-wind-speed spectra close to its maximum value, but it does not take into account the behaviour of the turbulent velocity fluctuations in the inertial subrange. In order to find a spectral form that describes both the low-wind-speed and the Kolmogorov − 2/3 trends, Mortarini and Anfossi (2015) proposed a linear combination of Eq. 5 with the Kaimal and Finnigan (1994) model for stable conditions,

$$ nF_{HW} \left( n \right) = A\left( {n/n_{0} } \right)/\left( {1 + A\left( {n/n_{0} } \right)^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} } \right), $$

(9)

where $n_{0}$ is the frequency at which the extrapolated inertial subrange meets the $nF_{HW} \left( n \right) = 1$ line and where $A = 0.164$ for both the three velocity components and the temperature. The meandering spectrum assumes the form

$$ F\left( {n,p,q} \right) = \alpha F_{LW} \left( {n,p,q} \right) + \beta \left( n \right)F_{HW} \left( n \right), $$

(10)

where $\beta$ is heuristically defined as

$$ \beta \left( {\tilde{n}} \right) = \left( {1 - 2\tilde{n} + \tilde{n}^{2} } \right)/\left( {1 + \lambda \tilde{n} + \tilde{n}^{2} } \right), $$

(11)

where $\tilde{n} = n/n_{max}$ and $\lambda$ is a shape parameter that takes into account the energy associated with the intermediate frequency range (i.e. between high and low frequencies), equalling 20 in stable conditions (Mortarini and Anfossi 2015). The parameter $\alpha$ in Eq. 10 is chosen to satisfy the normalization constraint

$$ \int_{0}^{\infty } {F\left( {n,p,q} \right) = 1,} $$

(12)

i.e.

$$ \alpha = 1 - \int_{0}^{\infty } {\beta \left( n \right)F_{HW} \left( n \right)dn} $$

(13)

where the integral is numerically solved using the adaptive quadrature method for each considered time series.

Finally, Eq. 10 can be written as

$$ \begin{aligned} F\left( {n,p,q} \right) & = 2p\left[ {\frac{1}{{p^{2} + \left( {q + 2\,\uppi\,n} \right)^{2} }} + \frac{1}{{p^{2} + \left( {q - 2\,\uppi\,n} \right)^{2} }}} \right] \\ & \quad + \,\frac{{\left( {1 - 2\frac{n}{{n_{max} }} + \left( {\frac{n}{{n_{max} }}} \right)^{2} } \right)}}{{\left( {1 + 20\frac{n}{{n_{max} }} + \left( {\frac{n}{{n_{max} }}} \right)^{2} } \right)}}\frac{{0.0164\frac{n}{{n_{0} }}}}{{1 + 0.0164\left( {\frac{n}{{n_{0} }}} \right)^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }} \\ \end{aligned} $$

(14)

which represents the spectra of the velocity components and of the temperature in the presence of submeso wavy motions.

After some algebra (for details, see Mortarini and Anfossi 2015), Eq. 14 can be normalized with the maximum of the low-frequency part of the spectrum (Eq. 7) to obtain,

$$ \begin{aligned} \tilde{F}\left( {\tilde{n},m} \right) & = \frac{{\tilde{n}F\left( {\tilde{n},p,q} \right)}}{{\left[ {nF_{LW} \left( n \right)} \right]_{max} }} = \alpha \frac{{2\tilde{n}\left( {1 + \tilde{n}^{2} } \right)}}{{m^{2} \left( {1 - \tilde{n}^{2} } \right)^{2} + \left( {1 + \tilde{n}^{2} } \right)^{2} }} \\ & \quad + \,\frac{{1 - 2\tilde{n} + \tilde{n}^{2} }}{{1 + 20\tilde{n} + \tilde{n}^{2} }}\frac{{0.515\frac{{\tilde{n}}}{\gamma }}}{{\sqrt {m^{2} + 1} \left[ {1 + 0.0164\left( {\frac{{\tilde{n}}}{\gamma }} \right)^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} } \right]}}, \\ \end{aligned} $$

(15)

where $\gamma = \left( {n_{0} /n_{max} } \right)$. The spectral quantity $\tilde{F}\left( {\tilde{n},m} \right)$ is useful in comparing averaged spectra and it does not depend on p and q separately but only on the ratio m.

The proposed model has been successfully tested for cases corresponding in an oscillatory behaviour in temperature field associated with horizontal meandering (Mortarini et al. 2016a).

Appendix 2: Wavelet Analysis and Cross-Spectral Indicators of Linear Gravity Waves

Wavelet analysis is a useful technique for investigating the intermittent and non-stationary structure of turbulence and its interaction with submeso motions in the ABL (Howell and Mahrt 1997; Cava et al. 2005, 2015, 2017; Viana et al. 2010; Durden et al. 2013; Sun et al. 2015b; Mortarini et al. 2018). The wavelet transform $Wf\left( {\lambda ,t} \right)$ of a function f(t) with finite energy is defined as

$$ Wf(\lambda ,t) = \int_{ - \infty }^{\infty } {f(u)\psi_{(\lambda ,t)} (u)du,} $$

(16)

where $\psi_{\lambda ,t} \left( u \right) = \left( {1/\lambda^{1/2} } \right)\psi \left( {\left( {u - t} \right)/\lambda } \right)$ is a family of functions depending on two parameters, a scale parameter $\lambda ( > 0)$ and a location parameter t. Changing the value of λ has the effect of dilating or contracting the function ψ (called the mother wavelet), i.e. of analyzing the function f(t) at different spatial scales, whereas changing t has the effect of analyzing the function f(t) around the point t.

The wavelet spectrum

$$ S = \left| {Wf\left( {\lambda ,t} \right)} \right|^{2} $$

(17)

gives information on the time evolution of the energy of the analyzed signal as a function of the resolved time scales (or frequencies). The wavelet cross-spectrum

$$ W_{fg} = Wf\left( {\lambda ,t} \right).*Wg\left( {\lambda ,t} \right) $$

(18)

allows the investigation of the common variability in frequency and time of two different signals (f, g); the cospectrum (C), i.e. the real part of $W_{fg}$, is proportional to the covariance between the two time series, whereas the quadspectrum (Q), i.e. the complex part of $W_{fg}$, represents the spectrum of the product of $f$ and $g$ shifted by 90° (Grinsted et al. 2004).

Cross-spectral statistics are very useful in diagnosing the linear nature of vertical wavy events detected in the measured series. In fact, in presence of linear waves, the lack of vertical diffusion produces fluctuations in vertical velocity and scalars 90° out of phase and, as a consequence, the quadspectral density is larger than the cospectral density in the wave source region (de Baas and Driedonks 1985).

For detailed discussions on wavelet theory and applications to the analysis of geophysical data, see Kumar and Foufoula-Georgiou (1997) and Torrence and Compo (1998).