In this thesis, we report on measurements of the energy dissipation rate and statistics of cloud droplets in the trade wind region. The rate of energy dissipation is one of the most fundamental characteristics of turbulence and is estimated from one-dimensional velocity time records.
Scope of the thesis
In general and regardless of the underlying mechanism, the rapid growth of cloud droplets is crucial for the initiation of precipitation. Both MPCK+ and mini-MPCK count and measure cloud points with the help of cloud point probes.
- Equations of motion
- Reynolds number and Nature of Turbulence
- Reynolds decomposition and Mean-flow Equations
- Homogeneous Isotropic Turbulence
- The Atmospheric Boundary Layer
- Random Error of Time-Average
The random error can be captured in terms of the estimation variance as given by . Equation (1.61) is the estimated variance of the mean velocity U due to a finite averaging window τ.
Cloud Droplet Growth due to Condensation
Consequently, the estimation variance of the mean speed (Eq. 1.61)), which can be regarded as a random error, causes a systematic underestimation of the variance. 1.63), we can also estimate the relative systematic error of the variance estimate, which is given by Eq. Due to the inverse relationship between the droplet diameter and the growth rate (in terms of dp), the droplet size distribution tends towards a narrower distribution with increasing average droplet diameter ¯dp around ¯dp .
Cloud Droplet - Turbulence Interaction
The relevant time scale for small flow rate changes is the Kolmogorov time scale τK, Eq. This chapter concludes with a discussion and perspective regarding a revised version of the mini-MPCK in Sec.
Max Planck Cloudkite+
Fluctuations in wind speed and air temperature are measured with a hot and cold wire at ~10 kHz. In summary, the MPCK+ is a mobile instrument that can be operated in remote regions of the world.
- Design requirements
- Mechanical Design
- Operating the mini-MPCK remotely
Assuming that the platform is always aligned with the prevailing wind direction, which is chosen to be the e1 direction, u = qu21+u22+u23 ≈ u1 in the reference frame of the platform if u1 ≪ u2,3. The dynamic pressure measurement error due to misalignment between the sensor and the mean wind direction is below 2.5% for angles of attack (or sideslip) between ±25°. It should be noted that the mini-MPCK is designed independent of the carrier platform.
Performance assessment during in-situ measurements
- In-field Operation aboard RV Meteor
- Flight Properties and Platform Motion
- Comparison to Radiosonde measurements
- Flow Velocity Measurements and Turbulence characterization . 50
Furthermore, the acoustic temperature measurement is the least affected temperature measurement by the mini-MPCK. In this section, we review the performance of the mini-MPCK in the field under EUREC4A.
Revert Running Average
Despite that, even more requirements will be fulfilled by the mini-MPCK 2.0., the mini-MPCK is an easy-to-operate and flexible instrument that can be operated not only on land-based towers, but also balloon-borne in very remote regions of the oceans. Assuming that the variability in T and ps is negligible on time scales below 100 ms, the wind speed can be obtained using the Bernoulli equation.
Air temperature from speed of sound
Invoking the ideal gas law for dry air and using the air temperature PSS8 T and the static pressure ps, the density of dry air is given by:.
Wind tunnel experiment
The micro-Max Planck Cloudkite
The GPS antenna (3) is attached to the outside of the Peli micro-MPCK box to receive a better GPS signal. The original photo is blurred where a person is partially shown to ensure privacy.
The mini-MPCK is near the outlet and points towards the mean wind direction.
The instantaneous energy dissipation field ϵ0(x, t) is highly intermittent with strong small-scale fluctuations [5, 42 and references therein], which is at the heart of the intermittency problem in turbulence. Apart from the instantaneous dissipation field ϵ0(x, t), the energy dissipation in a turbulent flow can be described statistically by either the local or global average energy dissipation rate, both of which are important. The local volume averages of the dissipation field converge to the global average energy dissipation rate ⟨ϵ⟩ for infinitely large average volumes.
3.2 we first define the central statistical quantities and the individual methods for estimating the energy dissipation rate in detail.
- On Averaging, Reynolds Decomposition and Taylor’s Hypothesis 70
- Estimating the Energy Dissipation Rate
- Simulations of homogeneous isotropic turbulence
- Variable Density Turbulence Tunnel
- Quantification of systematic and random errors
By complicating the estimation of the mean rate, random sweeping causes the mean energy dissipation rate to be consistently overestimated [137, 162]. This concludes the second-order statistics with respect to the velocity that we consider in the following to determine the average energy dissipation rate. Therefore, the instantaneous energy dissipation rate can only be defined from the velocity fluctuations , where in Eq.
Based on the inertial series scale of the nth-order longitudinal structure function, the mean energy dissipation.
Results and Discussion
- Verification of the analytical methods and a first insight into
- Validity of Taylor’s hypothesis and impact of random sweeping
- Probe Orientation
- Systematic and random errors due to finite averaging window
- Estimating the transient energy dissipation rate
3.3 we used the full time series so that the size of the averaging window is maximum. As mentioned above, the mean current direction U is considered to be the longitudinal direction of the current. Both the second-order structure function, Eq. 3.18), depend on the variance ⟨u′21⟩ of the longitudinal velocity time recording.
R , (3.44) where DLL(r;R) is the longitudinal second-order structure function evaluated over an averaging window of size R and under the assumption that the longitudinal autocorrelation function f(r) is well converged over the range of the averaging window .
Therefore, at small R3/2λ and R = 30L11, ⟨ϵI2⟩30L11 only scatters randomly around the global mean energy dissipation rate (with a 3% standard deviation of ⟨ϵG⟩30L11), which is why the correlation coefficient is low. In the case of finite turbulence intensities, ϵG, ϵS, and ϵI2 systematically overestimate the ground truth energy dissipation rate. With this we can estimate a coarse-grained energy dissipation rate within a predicted uncertainty as shown in Fig.
Only ϵG reliably estimates the transient energy dissipation rate ⟨ϵ⟩R, although it is most vulnerable to experimental imperfections/limitations.
Overview of M161 Flight 12
As shown by the zPSS8(t) barometric height time record of the micro-MPCK in Figs. The effect of the diurnal cycle on air temperature is shown by the time record of air temperature in Fig. In addition to atmospheric state parameters, such as static pressure ps and air temperature T, the micro-MPCK measured the relative wind in terms of longitudinal velocity u1(t) as a function of time (Fig. 4.1C).
Under the assumption of statistically stationary turbulence, a mean vertical velocity profile can be estimated to check the height at which the micro-MPCK leaves the wake of the RV Meteor.
Statistical Convergence in atmospheric turbulence
Therefore, for each τ for the entire time series, ⟨u′21⟩τ is evaluated according to the procedure in  where each averaging window τ corresponds to one realization of the flow. The shaded area is given by the standard error of the mean to capture statistical scatter. The systematic underestimation of the variance estimate is shown by the solid black line and given by Eq.
Therefore, I consider u1(t) statistically stationary if the changes of ⟨u′21(t)⟩τ according to the time shifts of the mean window τ are smaller than the random error of the variance estimate q2⟨u′21⟩2T11 /τ .
Bulk Turbulence Characteristics of M161 Flight 12
The resulting estimates of ϵI2 obtained from each DτLL(r) are cumulatively represented by the histogram of the energy dissipation rate estimates ϵI2. The estimation of the average energy dissipation rateϵI2 serves as the basis for estimating the Kolmogorov length scale ηK, Eq. 1.12), the longitudinal Taylor microscale λf, Eq. While the plausibility of the length scale has been demonstrated above, the significance of the Rλ estimate is assessed below.
In addition, due to the retrieval of the PSS8 velocity time measurement and the filtering in the frequency domain at 12 Hz (see Sect. 2.A), the local scaling exponent should not be trusted for scales below ~1 m.
Atmospheric Turbulence Characteristics of the Marine Boundary Layer 126
- Stability of the Boundary Layer
- Turbulence Characteristics in the Trade-wind Boundary Layer . 130
- Graphical Representation of Reynolds stresses
- Anisotropy in high-R λ flows
The barometric height zPSS8 is obtained from PSS8 and re-scaled by the top of the boundary layer (TBL). The top of the boundary layer (TBL) is obtained from radio soundings as described in Sec. TBL is the abbreviation for the top of the boundary layer and the averaging window τ = 30 min.
Thus, the estimation of the average energy dissipation rate based on averaging windows with τ = 30 min is justified.
In-situ Measurements of Trade-wind Cumuli
- Turbulence Characteristics of Trade-wind Cumuli
- Statistics of Cloud Droplets in Trade-wind Cumuli
- Cloud Droplet Statistics in Precipitating Cloud
The (local) mean energy dissipation rate ⟨ϵ⟩τ is obtained by means of windows τ = 60 s that compromise between the horizontal extent of the cloud and convergence of ⟨ϵ⟩τ. The spatial distribution of cloud droplets in trade wind cumuli is investigated as a function of the inter-particle distance ∆r. The total time series of the cloud droplet number concentration MSM89 Flight 12 is shown in Fig.
The estimates for the local average energy dissipation rate ⟨ϵI2⟩τ are obtained as explained in Sect.
Protocol for Validity Flag
Here, based on an averaging window of τ = 60 s, the σu′1 signal is median filtered with a kernel size of 30. However, there are still unphysical cloud droplet count concentrations in some localized events where the absolute cloud droplet count per time is significantly higher compared to other clouds of the same year. Furthermore, the algorithm explained above does not detect a non-physical wind speed measurement for a specific time interval during MSM89 Flight 12.
Further checking is needed in the case of MSM89 Flight 10 to verify high cloud drop numbers per time, but the velocity measurement is physical and the MPCK+ was far away from the ship.
Effect of Filtering
This suggests that fc = 0.5 Hz is in a regime that varies strongly with fc, which is unfavorable. However, taking into account that the references σu′1(0) and ϵI2(0) are also affected by the platform motions for cases (II) and (III), the relative comparison is not ideal. A complementary approach to investigate the filtering effect includes the fact that the average energy dissipation rate ⟨ϵ⟩ can be obtained by integrating the premultiplied energy spectrum E(κ) .
The helix acts as a weather vane that guides the movement of the tether-mounted mini-MPCK platform.
Since Rλ is very high in the atmosphere, small Rλ is not discussed further and the interested reader is referred to . 1 RMS fluctuation eu1(t),⟨ϵI2⟩τ estimation of average energy dissipation rate according to Eq. 3.15) with n= 2, ηK the Kolmogorov length scale, λf the Taylor microscale length, L11 the integral length scale and Rλ the Reynolds number on the Taylor microscale.
As mentioned earlier, the variance estimate and the energy dissipation rate estimate depend on the choice of the averaging window. Assuming that the error of the local mean energy dissipation rate is δϵ ≈22%, the relative error of St is 11%. It follows that the relative error of Sv is 5.5% due to the error of the local mean energy dissipation rate of δϵ ≈22%.
Direct Measurements of Radiant and Turbulent Flux Convergences in the Lower 1000 m of the Convective Boundary Layer”.