Average Path Profile of Atmospheric Temperature and Humidity Structure Parameters from a Microwave Profiling Radiometer

The values of the key atmospheric turbulence parameters (structure constants) for temperature and water vapor, i.e., CT2, and CQ2, are highly dependent upon the vertical height within the atmosphere thus making it necessary to specify profiles of these values along the atmospheric propagation path. The remote sensing method suggested and described in this work makes use of a rapidly integrating microwave profiling radiometer to capture profiles of temperature and humidity through the atmosphere. The integration times of currently available profiling radiometers are such that they are approaching the temporal intervals over which one can possibly make meaningful assessments of these key atmospheric parameters. These integration times, coupled with the boundary effects of the Earth’s surface are, however, unconventional for turbulence characterization; the classical Kolmogorov turbulence theory and related 2/3 law for structure functions prevalent in the inertial sub-range are no longer appropriate. An alternative to this classical approach is derived from first principles to account for the nuances of turbulent mechanics met with using radiometer sensing, i.e., the large-scale turbulence driven by the various possible boundary conditions within the buoyancy sub-range. Analytical expressions connecting the measured structure functions to the corresponding structure parameters are obtained. The theory is then applied to an experimental scenario involving radiometric profile measurements of temperature and shows very good results.


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Introduction
The atmospheric turbulence metrics inherent in the definitions of the structure constants of passive additives such as temperature, 2 T C , and water vapor (humidity), 2 Q C , are not only important in the assessment of the tropospheric turbulence field but also in the assessment of the radio and optical refractive index field in the consideration of the propagation of electromagnetic waves. Temperature and water vapor are the major components that determine the prevailing refractive index field (characterized by the refractive index structure constant, 2 n C ) and their statistical evaluation is a prerequisite for the performance of image and communications systems that must rely on electromagnetic wave transmission through the atmosphere. Values of these structure parameters are functions of height above the Earth's surface and a comprehensive description of their behavior must include such variation of their values along a vertical path profile through the atmosphere.
Within the confines of classical Kolmogorov turbulence theory, the value of 2 T C can be determined through the operational definition provided by the 2/3 law, that is, 2 〉 is the temperature structure function, the value of which is easily determined through measurement of the temperature difference across the spatial separation d and the subsequent temporal average of its difference is squared. Similarly, the same consideration holds for water vapor. The subject of the present work is the exploratory analysis of a measurement technique that can capture vertical profile values of temperature and water vapor in the atmosphere. In particular, the remote sensing method suggested and described in this work makes use of a rapidly integrating microwave profiling radiometer (Radiometrics Corp. MP-3000A) to capture these profiles. The essence of the method is to capture two such profiles consecutively measured over a time interval of ∆t, the integration time of the radiometer. Then, via the application of the Taylor frozen-flow hypothesis, one forms a single realization of the structure function across the induced spatial separation U t ∆ , where U is a vertical profile of the average wind velocity. Upon ensemble averaging of such measurements, a value of the corresponding structure parameter can be obtained along the vertical profile. The integration time of this particular radiometer is on the order of 30 to 40 s and is such that it approaches the temporal intervals over which one can possibly make meaningful measurements of some key atmospheric parameters. This query, of course, implicitly assumes many things. The major ones are as follows: (1) a relationship similar to the 2/3 law, but applicable to the largescale turbulence phenomena involved, can be identified and used to do the actual calculation of the structure function values, (2) applicability of the frozen-flow hypothesis over the integration period ∆t, and (3) the resolution requirements placed upon the radiometer to discern the difference values of temperature and water vapor typical of atmospheric scenarios. The most important aspect of these assumptions is the determination of the relationship that replaces the 2/3 law that only holds under idealistic conditions. That is, one must establish a general function F(d) such that, in the case of temperature, for example, 2 The form of this function will be strongly dependent on boundary conditions such as shear flow, buoyancy, stability, ∆t, etc., and of course, should reduce to F(d) ≈ d 2/3 in the Kolmogorov case. The determination of this function will dominate this work as it is key to the entire calculation process that results in the structure parameter values. Also, the frozen-flow hypothesis will be extended to prevail over the relatively large values of ∆t, which too will enter into the determination of F(d) if it is deemed a significant factor. Section 2.0 begins with a brief review of the theory of the large-scale atmospheric turbulence spatial spectrum near the surface of the Earth. Here, "large-scale" refers to characteristic turbulence sizes on the order of l U t ∆  ; for a nominal horizontal average wind speed of U ≈ 5 m/s and radiometer integration time of ∆t ≈ 40 s, l ≥ 200 m. Since the radiometer measurements are made at the Earth's surface, it is required to take into account the surface or boundary layer effects on the formation of such large-scale turbulent fluctuations in the presence of the vertical gradients of velocity, temperature, and water vapor within the atmosphere; the well-known Kolmogorov spectrum for the inertial subrange of isotropic turbulent fluctuations cannot be directly applied. Hence, the Kolmogorov theory must be transcended to account for these effects in a generally stratified atmosphere within the buoyancy subrange. Although such phenomena have been considered earlier (Ref. 1), a self-contained theory is given in Section 2.0 that is general enough to quantitatively apply to many turbulence scenarios.
Here, stable as well as unstable cases in an atmosphere with vertical gradients of temperature and velocity (shear) are considered. A composite spatial spectrum for both the inertial and buoyancy subranges is then given as a function of atmospheric conditions and the corresponding structure functions for temperature and humidity are derived. The form of the model connecting the structure functions to those determined from the radiometric measurements is then finally derived in Section 3.0, thus determining the function ( ) F U t ∆ . Due to the relatively large value for ∆t, a Fourier-Stieltjes treatment is employed, shown in Appendix A, that transcends the usual use of the Taylor frozen-flow hypothesis in the event that it no longer holds in this temporal region. It is established, however, that the hypothesis does hold well for the large turbulent inhomogeneity sizes for which this sensing technique depends. The demands placed on the resolution requirements for temperature and water vapor of the radiometer are then found. Finally, in Section 4.0, experimental demonstration of the remote sensing method will be given.
A preliminary study of this problem was undertaken in Reference 2 where a composite spectrum for the inertial and buoyancy subranges was advanced and used. However, it provided a poor approximation for the desired model of the turbulence scenario. This circumstance is rectified in the present work. Much of the detail of its derivation from first principles is retained here for completeness. It must be noted that many ranges of atmospheric turbulent spectra can be obtained and quantitatively connected to atmospheric parameters from this analysis. This development was necessary to obtain a firm theoretical basis for this type of remote sensing technique.

The Spatial Spectrum of Turbulent Fluctuations in Thermally Stratified Atmosphere With Shear Flow
In this section, expressions for the spatial spectrum of the combined small-and large-scale turbulent flows will be developed and analyzed.

Development of Spectral Model for Atmospheric Turbulence for Large Scales
The incompressible turbulent flow within the atmosphere that governs the spatial and temporal evolution of the velocity field V  is given by the Navier-Stokes equation; employing the Boussinesq approximation and assuming a constant dynamic viscosity, µ, one has (Ref. 3): for i = 1,2,3, ρ0 and ρ are the mean and instantaneous density, respectively, P is the pressure and g3 is the gravitational acceleration along the vertical 3 ( ) x z ≡  axis. Additionally, the atmospheric temperature field T, which is the source of density fluctuations, is given by (Ref where cP is the heat capacity at constant pressure and µT is the thermal conductivity of air. A similar equation holds for the water vapor field Q. In this report, only the temperature field will be considered with the proviso that the final results will hold for Q (so long as T and Q act as passive additives). Following the detailed procedure given in Reference 4, Equations (1) and (2) are statistically analyzed to give equations involving the Fourier spectra of the velocity and temperature fluctuations: Here, k is the wavenumber, F(k) and FTT(k) are the energy transfer spectra due to the distortion of fluctuation gradients of, respectively, velocity fluctuations and temperature fluctuations, φ13(k) is the spectrum of the energy due to the work of velocity fluctuations from Reynolds stresses against the mean shear, φ3T(k) is the spectrum of the energy due to the work of temperature fluctuations transferred by vertical heat flux against the temperature gradient, and φ(k) and φTT(k) are the spectra of, respectively, turbulent energy fluctuations and temperature fluctuations. The mean atmospheric temperature T and velocity U are, in general, both functions of the height coordinate z within the atmosphere, that is, the atmosphere is stratified. Finally, v ≡ µ/ρ0 is the kinematic viscosity, vT ≡ µΤ /(ρ0cP) is the thermal diffusivity, and / g T β ≡ is the buoyancy parameter and g is the gravitational acceleration. Equations (3) and (4) can be integrated to give the more familiar form: where the total dissipation of turbulent energy by viscosity is and the total dissipation of temperature fluctuations by thermal conductivity is The point of this development is to obtain from Equations (5) and (6) functions for the turbulent velocity spectrum φ(k) and, most importantly, the temperature fluctuation spectrum φTT(k) and associate them to well-defined parameters that characterize the various atmospheric conditions, which can prevail during a radiometer measurement. Once the spectrum φTT(k) is obtained, it is a simple matter to calculate the associated temperature (or humidity) structure function and apply it to the radiometer profiles to determine the related parameter, 2 T C . However, at this point, the classical problem well known in turbulence theory is met, namely, due to the nonlinearity of the equations obtained (the source of which is the basic nonlinearity of the Navier-Stokes equations), the number of unknowns is larger than the number of equations, that is, a closure problem prevails. Within the spectral approach considered here (as opposed to the statistical correlation approach), further statistical assumptions involving the turbulent energy spectral transfer functions need to be employed, which allow connections of them to φ(k) and φTT(k). This is thoroughly discussed in Reference 4. See also Reference 5 (Sec. 17) for a comprehensive treatment.
The method is essentially as follows. Following Heisenberg's approach (Ref. 6), the φ(k) given in the third term of Equation (5) is written in which η(k) is the kinematic eddy viscosity. For purposes of this development, the expression used for η(k) will not be that given in Reference 7 but one that is more appropriate for the large Prandtl numbers (i.e., viscous diffusion exceeding that of thermal diffusion) typical of atmospheric turbulence (Ref. 7): where γ is a numerical constant on the order of unity. The idea behind the model of Equations (9) and (10) is that the transfer of energy from fluctuations of wavenumbers less than k to fluctuations of wavenumbers larger than k can be taken as occurring through the viscosity that exists between the fluctuation eddies working on the turbulent vorticity formed in the interval 0 to k. This viscosity can be modeled as the integral effect of fluctuation eddies with wavenumbers larger than k acting on eddies with wavenumbers less than k. The functional form of Equation (10) (9) and (10): A similar argument can be applied to the last term of Equation (6) where b is the ratio of vT to v of the fluctuation eddies; it too is on the order of unity. The same methodology can be applied to connect the spectra φ13(k) and φ3T(k) to φ(k) and φTT(k). To do this, one must account for the interactions between the gradients of the U and T with the overall turbulent field (Ref. 4). One must also consider the level of interaction that the velocity field has on the temperature gradients within the stratified atmosphere; such interaction concepts were first put forward by Tchen (Ref. 1) (using the term "resonance") who considered the similar problem of deriving a turbulence spectrum perturbed by boundary effects. Here, the case of the strong interaction is considered. For the model of η(k) given by Equation (10) and based on these considerations as well as those of the dimensionality of the quantities involved, one has (see Ref. 4 for details) and ( ) Substituting Equations (11) to (14) into Equations (5) and (6) yields where the upper sign on Equation (15) is for the case / 0 dU dz > and the lower sign for / 0 dU dz < . Similarly, the upper sign in Equation (16) is for / 0 dT dz > (which defines the case of stable stratification of the atmosphere) and the lower sign for / 0 dT dz < (which defines the case of unstable stratification of the atmosphere). Equations (10), (15), and (16) concatenate everything that goes into the determination of the φTT(k) spectrum, within the bounds of the assumptions that enter into the closure approximations that allow Equations (11) to (14) to be written. The method of solution for φTT(k) using the general model afforded by these relations will be the subject of a future publication. A special case of these equations will be used here to find analytical solutions for φTT(k) appropriate for the establishment of analytical connections between the measured temperature (or humidity) structure functions derived from the radiometer output and the structure parameter 2 2 (or ) T Q C C . To this end, since large-scale turbulence is being considered, one can ignore the contribution of molecular diffusion effects in the evolution of the spectra thus allowing the first terms on the right sides of Equations (15) and (16) to be dropped. Doing so yields Finally, converting to dimensionless variables defined by Equations (17) and (18) become ( ) Equations (22) and (23) can be considered simply as simultaneous algebraic equations to be solved for L and M, both as functions of K(x). Then using Equation (27) with these solutions will yield a set of parametric equations involving the spectra as well as the function K(x) along with, of course, the parameters ΓU and ΓT that characterize the atmospheric conditions. Within various combinations of limits of ΓU and ΓT, these parametric equations can be first solved for K(x) and then for Φ and ΦTT using the additional relation from Equation (26), that is:

Solutions of Equations of Spectral Model
Solving Equations (22) Differentiating the first relation of Equation (29) with respect to x and using Equation (27) gives, after a bit of manipulation: Similarly, differentiating the first relation of Equation (30) with respect to x gives, again after some algebraic manipulation: Using these relationships, several combinations and permutations of atmospheric scenarios can be considered. This will form the subject of a future publication. For purposes of this exposition, these equations will now be used to derive analytical expressions for the spectra in two extreme cases: ( ) ( ) which is the result for the inertial subrange that defines this case. Putting this result into Equation (31) then gives for the attendant temperature spectrum: which defines the buoyancy subrange for a stratified atmosphere.

Height-Dependent Spectrum for Both Buoyancy and Inertial Subranges
Thus, the temperature spectrum in the case of no stratification or shear, that is, one which is expected to prevail in the atmosphere away from the surface layer, is, returning to dimensional variables using Equations (19) to (21): and in the opposite case of shear and stratification, that is, one which is expected to prevail in the atmosphere close to the boundary surface: In the general case intermediate to these, one would expect the temperature spectrum to transition from that given by Equation (40) to that given by Equation (39) as one proceeds vertically up through the atmosphere from the Earth's surface to above the boundary layer. Also, Equation (40) will prevail over the large spatial separations between two temperature profiles that are involved with the present radiometer remote sensing technique, that is, over small k; Equation (39) governs the spectrum over small spatial separations, that is, large k. A composite expression for the temperature spectrum that approaches Equation (39) as k → ∞ and approaches Equation (40) as k → 0 is desired. Unlike the method adopted in Reference 2, the attempted combination of the two turbulence regions considered here begins with the virtual viscosity functions KT (x) and KU (x). Such a combination that reflects the limiting behavior in both the x and ΓU domains defining these particular regions is given by This becomes a complicated expression when considering all the ranges of values that HT can assume for both dT/dz < 0 and dT/dz > 0, etc. A straightforward but detailed analysis of Equation (42), which will not be reproduced here, yields for cases (i) and (ii): where the condition 2 2 U T Γ > Γ must prevail in the limit as ΓT → ∞. Using Equations (19) to (21) and simplifying, Equation (43) finally becomes ( ) where two characteristic spatial frequencies appear defined by 3 and 2 2

Evaluation of Characteristic Spatial Frequency Coefficients for In Situ Applications via Similarity Theory
Similarity theory can be applied to obtain numerical expressions for the vertical profiles of the coefficient B as well as the prevailing values of kU and kT using their relations to the fundamental atmospheric parameters given by Equations (45) and (46). These profile estimates are very helpful (but not required) to use with the corresponding profiles of temperature and water vapor from the radiometer. The basic idea is this: in order to apply the master equation of this remote sensing method, which will be derived below directly from Equation (44)

Frozen-Flow Hypothesis and Relating Structure Parameters to Measured Structure Functions-Transcending 2/3 Law
With the spatial spectrum now established that attempts to cover the regions of turbulent activity met within the atmospheric boundary layer in the application of the remote sensing method considered here, it now seems to be a straightforward matter to form the expression for the corresponding structure function (Refs. 5 (Sec. 13) and 11). In the case of a spatial separation d between two spatial points, one has for the temperature structure function: so long as ∆t is small enough to assure that the evolution of the turbulent field does not occur. Using the profiling radiometer employs ∆t ~ 30 to 40 s so the hypothesis of frozen flow may be in question. Appendix A presents a first-principles derivation of the modifications that are induced in the use of Equation (48) over large ∆t with the result that Substituting the spectrum of Equation (44) into Equation (49) and performing the required integrations yield analytical expressions involving combinations of incomplete Γ functions and complex exponentials. Converting these functional combinations into corresponding confluent hypergeometric functions Ψ(a;b;z) for ease of numerical evaluation, one has ( ) To be sure, in the limit:

8.04
which recovers the 2/3 law as well as establishes that the temperature structure parameter which corresponds to the definition of B in Equation (46). However, the second member of Equation (50) that corrects for possible deviations from the frozen-flow hypothesis can be found to be negligible for spatial frequencies kt < 1/(υ∆t); for 0.1U υ  with 5 m/s U  and ∆t = 40 s, kt < 0.05 m -1 . This is the point of demarcation shown in Figure 1 where the spectrum becomes dominated by the effects of the large turbulent inhomogeneity sizes to which this sensing method applies. Hence, one finally establishes the master equation of this proposed remote sensing method:

Experimental Verification of Remote Sensing Technique
A preliminary experimental demonstration of the method advanced in this report is provided by the use of a profiling radiometer (Radiometrics Corp. MP-3000A) having 35 calibrated channels with a 1.1 s integration time per channel giving ∆t ≈ 40 s. The bandwidth per channel is 300 MHz in the 22.0 to 30.0 GHz and 51.0 to 59.0 GHz (K and V) bands. The temperature resolution was 0.1 K. The measurements were taken in January 2013 at the NASA Tracking and Data Relay Satellite System (TDRSS) ground terminal site located at White Sands, New Mexico, with the radiometer pointed to zenith. The dataset comprised 2,100 temperature profiles taken over a 24-h period. The vertical heights of the profiles were discretized over 50 m intervals up to a maximum height of 10 km. Only temperature profiles were considered here. Unfortunately, specific atmospheric conditions during the radiometric data compilation were not available during the time the dataset was obtained. Specific considerations and details of the discretization procedures required as well as application of moving averages to the raw data appear in an earlier publication (Ref. 2). Two major considerations must be noted. First, the need for discrete wind velocity profiles that capture local prevailing conditions can be obtained using available methods (Ref. 13). Second, the required finite integration time of the radiometer restricts the method to apply beginning at a minimum height above the surface. The larger the value of ∆t, the larger the minimum height hmin is above the surface below, which the calculated structure parameters cannot be resolved. Assuming isotropic behavior of the turbulent inhomogenieties that the method can discern, one can simply place this minimum height at the value min h U t = ∆ . Of course, contributions of the atmosphere below this height that determine the value of the structure parameters is significant. The experimental derivation of the gradient Richardson number at the radiometer site, as discussed earlier, concurrent with the profile measurements, will secure the surface values and profiles of the structure parameters up to hmin. Figure 4 displays the result of obtaining 143 values of ( ) T D U t ∆ from the temperature profile dataset and using these derived values in Equation (52) to find 143 corresponding profiles of 2 T C . The averaging required to form the structure function values were obtained from 10-min moving averages of the raw differences of adjacent temperature profiles. Since atmospheric measurements were not taken to secure the prevailing values of kt and kU, nominal constant magnitudes of kt = 0.9 m -1 and kU/kt = 1.0 m -1 typical of a stable atmosphere were selected. Wind profiles were created using the method described in Reference 13 in conjunction with historical highresolution radiosonde data compiled over 3 years at the measurement location. A principle component analysis was then applied to the wind data to obtain a statistical model for U as a function of height. (Such a statistical wind profile model can be obtained using Ref. 13 for any location with a long-term wind profile database.) Figure 4 shows the minimum height limitation along the abscissa as well as instances where values smaller than min 2 2 2 3 0.003 K m T C = are obtained. Unfortunately, no concurrent, independent atmospheric measurements were being made to secure the 2 T C profiles for comparison. However, the morphological behavior of those shown in Figure 4 are the same as those obtained by other methods.

Discussion
The results obtained and displayed in Figure 4 are certainly encouraging enough to provide motivation for a controlled experimental verification of the remote sensing method. To this end, profiling radiometer data for both temperature and water vapor (which was ignored in this report) must be captured. In addition, corresponding simple atmospheric measurements are made to characterize the gradient Richardson number, from which similarity theory can be used, as described in this report, to provide estimates of the profile values of kU and kT needed to aid in the use of the master Equation (52) from which 2 2 and T Q C C profiles can be obtained from the radiometer derived measurements that determine ( ) T D U t ∆ and ( ) Q D U t ∆ . These atmospheric measurements can also augment other methods used to obtain 2 2 and T Q C C profile data to provide quasi-independent verification. Also, the use of in situ radiosonde measurements concurrent with the formation of the radiometer data to provide yet another independent verification would be invaluable. Work is now progressing toward these goals.

Conclusions
A remote sensing method using a profiling microwave radiometer to assess vertical path profiles of temperature and water vapor structure parameters has been proposed and experimentally shown to be promising. The ability to accomplish this task relied on two issues: (1) the integration time of profiling radiometers have become small enough for potential consideration of atmospheric turbulent field assessment through the actual measurement of the associated structure functions for the passive additives and (2) the development of a theoretical basis to provide the turbulent fluctuation spectra that is encountered using such unconventionally large measurement times, that is, large-scale turbulence driven by the various possible boundary conditions; one cannot expect Kolmogorov theory to hold that applies only in the inertial subrange. The resulting spectral theory that was obtained seems to have the flexibility to treat many combinations of atmospheric turbulence conditions. Since large-scale phenomena are considered, effects of molecular viscosity are ignored. For purposes of providing a basis for the radiometer remote sensing technique, two disparate regions of atmospheric turbulence activity were chosen, namely, no atmospheric stratification or shear and significant stratification and shear. A composite turbulence spectrum was then obtained from which a general scaling law was derived to replace the specialized 2/3 law. Thus, the structure parameter profiles of 2 2 and T Q C C (respectively, the constants for temperature and water vapor) can be obtained by forming the structure functions of the respective quantities from the radiometer measurements. The turbulent dynamics theory that was developed also was coupled to similarity theory to provide that capability to perform simple in situ temperature and water vapor gradient measurements to capture the relative magnitudes of the coefficients that appear in the spectrum.
Although the rather quick experimental verification of this technique is promising, more comprehensive verification must be done. Additionally, the use of a composite spectral representation from the theory developed here that employs other atmospheric turbulent conditions intermediate to those used here should also be considered. The issues of just what spectral form to use will be settled through careful atmospheric characterization concurrent with the radiometer measurements.