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- Author or Editor: Tzvi Gal-Chen x

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## Abstract

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## Abstract

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## Abstract

This paper develops a theory for the estimation of virtual temperature from remote measurements of (i) emitted thermal radiation by microwave and infrared radiometers and (ii) horizontal winds by Doppler radars (or lidars). The problem of obtaining a temperature profile from measured radiances and an inversion of the radiative transfer equation is known to be ill posed/conditioned. For a meaningful inversion additional information is needed. In this work we utilize two sources of information and two physical relationships which are: 1) measured radiances at various frequencies; 2) the radiative transfer equation at various frequencies; 3) the measured horizontal wind, and 4) the horizontal equations of motions. We then look for a virtual temperature profile that will (in the least-square sense) satisfy both the horizontal equations of motions and the radiative transfer equations. The resulting calculus of variations problem, after suitable manipulations, is then reduced to solving at each horizontal level a Poisson equation for the virtual temperature. The input parameters are the measured horizontal winds from which, using objective analysis techniques, one can derive gradients and various nonlinear products of the wind, the measured radiances and its derived gradients and the kernels (at various frequencies) of the radiative transfer equation. The sensitivity of the solution to uncertainties in the vertical velocities and to objective analysis errors is also examined. The inclusion of fluid dynamical considerations also allows for the allows of components of the virtual temperature profile which make no contribution to the measured radiances.

## Abstract

This paper develops a theory for the estimation of virtual temperature from remote measurements of (i) emitted thermal radiation by microwave and infrared radiometers and (ii) horizontal winds by Doppler radars (or lidars). The problem of obtaining a temperature profile from measured radiances and an inversion of the radiative transfer equation is known to be ill posed/conditioned. For a meaningful inversion additional information is needed. In this work we utilize two sources of information and two physical relationships which are: 1) measured radiances at various frequencies; 2) the radiative transfer equation at various frequencies; 3) the measured horizontal wind, and 4) the horizontal equations of motions. We then look for a virtual temperature profile that will (in the least-square sense) satisfy both the horizontal equations of motions and the radiative transfer equations. The resulting calculus of variations problem, after suitable manipulations, is then reduced to solving at each horizontal level a Poisson equation for the virtual temperature. The input parameters are the measured horizontal winds from which, using objective analysis techniques, one can derive gradients and various nonlinear products of the wind, the measured radiances and its derived gradients and the kernels (at various frequencies) of the radiative transfer equation. The sensitivity of the solution to uncertainties in the vertical velocities and to objective analysis errors is also examined. The inclusion of fluid dynamical considerations also allows for the allows of components of the virtual temperature profile which make no contribution to the measured radiances.

## Abstract

Recent suggestions to use the full divergence equation and nearly continuous (in time) wind observations to retrieve the geopotential and temperature of large-scale and mesoscale (but still hydrostatic) phenomena are examined. By extrapolating from results obtained in small-scale meteorology the following is suggested: 1) boundary conditions for the normal derivative of temperature (or the geopotential) can be obtained from the wind data itself and 2) objective criteria can be devised to indirectly test the quality of the results and also to decide whether Neumann (i.e., a case in which the normal derivative is specified) or Dirichlet (i.e., the function itself is specified) type boundary conditions should be enforced.

## Abstract

Recent suggestions to use the full divergence equation and nearly continuous (in time) wind observations to retrieve the geopotential and temperature of large-scale and mesoscale (but still hydrostatic) phenomena are examined. By extrapolating from results obtained in small-scale meteorology the following is suggested: 1) boundary conditions for the normal derivative of temperature (or the geopotential) can be obtained from the wind data itself and 2) objective criteria can be devised to indirectly test the quality of the results and also to decide whether Neumann (i.e., a case in which the normal derivative is specified) or Dirichlet (i.e., the function itself is specified) type boundary conditions should be enforced.

## Abstract

An algorithm is proposed, whereby the combined use of the equations of cloud dynamics, and the observed wind, will permit a unique determination of the density and pressure fluctuations. The algorithm has several unique features. First, it involves the use of only the momentum equations without resorting to any thermodynamical (microphysical) parameterizations. Second, there is no need to make any artificial assumptions about the boundary conditions. Instead, the algorithm determines (from wind observations) its own optimal boundary conditions. This latter feature is crucial for severe storm observations where very often the boundary conditions are not those of the larger scale environment.

The viability of the method is tested with data generated by a numerical model. From these data the wind and its time derivative are estimated and used to calculate the density and pressure fluctuations. These calculations are then compared to the originally given density and pressure. The sensitivity of the method to various observational errors is assessed by inserting realistically simulated errors into the “observed” kinematics. The principal findings are as follows: 1) good results can be obtained, if the filtering (noise removal) technique takes into account subgrid-scale processes; 2) at least for some applications, doubt is cast on the validity of the local steady-state hypothesis, which is now commonly used. It is argued, nevertheless, that the nonsteadiness can be readily evaluated from observations.

## Abstract

An algorithm is proposed, whereby the combined use of the equations of cloud dynamics, and the observed wind, will permit a unique determination of the density and pressure fluctuations. The algorithm has several unique features. First, it involves the use of only the momentum equations without resorting to any thermodynamical (microphysical) parameterizations. Second, there is no need to make any artificial assumptions about the boundary conditions. Instead, the algorithm determines (from wind observations) its own optimal boundary conditions. This latter feature is crucial for severe storm observations where very often the boundary conditions are not those of the larger scale environment.

The viability of the method is tested with data generated by a numerical model. From these data the wind and its time derivative are estimated and used to calculate the density and pressure fluctuations. These calculations are then compared to the originally given density and pressure. The sensitivity of the method to various observational errors is assessed by inserting realistically simulated errors into the “observed” kinematics. The principal findings are as follows: 1) good results can be obtained, if the filtering (noise removal) technique takes into account subgrid-scale processes; 2) at least for some applications, doubt is cast on the validity of the local steady-state hypothesis, which is now commonly used. It is argued, nevertheless, that the nonsteadiness can be readily evaluated from observations.

## Abstract

^{2}/∂

*t*

^{2}(

*v*

_{r}r)]

^{2}is a minimum. Here

*v*is the radial velocity,

_{r}*t*is time and

*r*is the distance of the observed point from the radar. In the multiple Doppler case, it is shown that in a frame of reference moving with the advection speed,

*t*

*v*

_{r}

^{(1)}

*r*

^{(1)}

*t*

*v*

_{r}

^{(2)}

*r*

^{(2)}

^{2}

## Abstract

^{2}/∂

*t*

^{2}(

*v*

_{r}r)]

^{2}is a minimum. Here

*v*is the radial velocity,

_{r}*t*is time and

*r*is the distance of the observed point from the radar. In the multiple Doppler case, it is shown that in a frame of reference moving with the advection speed,

*t*

*v*

_{r}

^{(1)}

*r*

^{(1)}

*t*

*v*

_{r}

^{(2)}

*r*

^{(2)}

^{2}

## Abstract

Numerical computations of collision efficiencies and certain coalescence parameters are presented for droplet pairs with Reynolds numbers up to 104. This corresponds to radii (for 900 mb, 0C) of up to 300 μ. The superposition principle was used, and the flow fields were computed from the complete nonlinear, time-dependent Navier-Stokes equations, using a modified Rimon's method. The collision efficiencies are mostly geometrical for intermediate ratios of the drops radii, 0.3 < *p* < 0.7, and exceed the geometric line considerably for high *p* values. The discussion centers on the validity of the results, as well as their meaning in terms of cloud physics.

## Abstract

Numerical computations of collision efficiencies and certain coalescence parameters are presented for droplet pairs with Reynolds numbers up to 104. This corresponds to radii (for 900 mb, 0C) of up to 300 μ. The superposition principle was used, and the flow fields were computed from the complete nonlinear, time-dependent Navier-Stokes equations, using a modified Rimon's method. The collision efficiencies are mostly geometrical for intermediate ratios of the drops radii, 0.3 < *p* < 0.7, and exceed the geometric line considerably for high *p* values. The discussion centers on the validity of the results, as well as their meaning in terms of cloud physics.

## Abstract

The kinematic and dynamical properties of the convective planetary boundary layer (CBL) with shear are studied using single Doppler radar measurements. The data were collected using single K-band (0.87 cm) Doppler radar operated by the National Oceanic and Atmospheric Administration Wave Propagation Laboratory. An analysis technique has been developed and tested that is an extension of the velocity azimuth display (VAD). The technique and its associated error analysis are described in detail. Profiles of the mean wind, vertical velocity, associated momentum fluxes, and variances are estimated. In agreement with the classical picture of the CBL, the profile of the vertical velocity variance achieves a relative maximum magnitude in the lower part of the mixed layer. Large magnitudes of vertical flux of horizontal momentum are found at the top of the mixed layer, while countergradient momentum fluxes are found in part of the CBL. The second moment of the Doppler spectrum is utilized, so that the momentum transport by eddies smaller than radar resolution volume is included. In most cases and contrary to most large eddy simulations studies, the effects of the subresolution momentum fluxes on the mixed layer are not negligible.

The longitudinal power spectrum of the horizontal velocity near the surface has also been calculated using the Fourier transform techniques. Traditional κ^{−5/3} spectra at the inertial subrange are obtained from which, using standard techniques, the turbulent kinetic energy dissipation rate has been estimated. To test the validity of the single Doppler analysis techniques, the derived turbulence statistics are compared with the results from other measurements.

## Abstract

The kinematic and dynamical properties of the convective planetary boundary layer (CBL) with shear are studied using single Doppler radar measurements. The data were collected using single K-band (0.87 cm) Doppler radar operated by the National Oceanic and Atmospheric Administration Wave Propagation Laboratory. An analysis technique has been developed and tested that is an extension of the velocity azimuth display (VAD). The technique and its associated error analysis are described in detail. Profiles of the mean wind, vertical velocity, associated momentum fluxes, and variances are estimated. In agreement with the classical picture of the CBL, the profile of the vertical velocity variance achieves a relative maximum magnitude in the lower part of the mixed layer. Large magnitudes of vertical flux of horizontal momentum are found at the top of the mixed layer, while countergradient momentum fluxes are found in part of the CBL. The second moment of the Doppler spectrum is utilized, so that the momentum transport by eddies smaller than radar resolution volume is included. In most cases and contrary to most large eddy simulations studies, the effects of the subresolution momentum fluxes on the mixed layer are not negligible.

The longitudinal power spectrum of the horizontal velocity near the surface has also been calculated using the Fourier transform techniques. Traditional κ^{−5/3} spectra at the inertial subrange are obtained from which, using standard techniques, the turbulent kinetic energy dissipation rate has been estimated. To test the validity of the single Doppler analysis techniques, the derived turbulence statistics are compared with the results from other measurements.

## Abstract

A single Doppler radar analysis scheme is developed, and three-dimensional wind fields am retrieved from single Doppler radar reflectivity and radial velocity fields. The retrieval is based on two assumptions: 1) the Lagrangian conservation of the radar reflectivity and 2) the steadiness of the eddy structures in the wind field. To least violate these assumptions, a moving frame of reference is found where (in the least square sense) the observations are as stationary as possible. Multiple time levels of observations are used to avoid ill-conditioned computations. The retrieval equations, that is, the conservation equation of reflectivity and the relationship between the total wind and its radial component for several time levels, form a simple linear system. This linear system is overdetermined with respect to the three unknown Cartesian components *u*, *v*, and *w* of the vector wind. Thus *u*, *v*, and *w* are solved in the least square sense. Dual Doppler radar analyses are performed to provide verifications for the single-Doppler retrievals. The results show very good agreement between the wind fields from single-Doppler retrievals and the dual-Doppler analyses. The important findings from various experiments include that 1) the weighting of each term in the cost function is crucial to the retrieval accuracy; 2) performing the retrievals in the moving frame improves the results significantly; and 3) proper filtering in space and time can reduce errors in retrieved wind fields. Two independent cases are studied to test the robustness of the scheme.

## Abstract

A single Doppler radar analysis scheme is developed, and three-dimensional wind fields am retrieved from single Doppler radar reflectivity and radial velocity fields. The retrieval is based on two assumptions: 1) the Lagrangian conservation of the radar reflectivity and 2) the steadiness of the eddy structures in the wind field. To least violate these assumptions, a moving frame of reference is found where (in the least square sense) the observations are as stationary as possible. Multiple time levels of observations are used to avoid ill-conditioned computations. The retrieval equations, that is, the conservation equation of reflectivity and the relationship between the total wind and its radial component for several time levels, form a simple linear system. This linear system is overdetermined with respect to the three unknown Cartesian components *u*, *v*, and *w* of the vector wind. Thus *u*, *v*, and *w* are solved in the least square sense. Dual Doppler radar analyses are performed to provide verifications for the single-Doppler retrievals. The results show very good agreement between the wind fields from single-Doppler retrievals and the dual-Doppler analyses. The important findings from various experiments include that 1) the weighting of each term in the cost function is crucial to the retrieval accuracy; 2) performing the retrievals in the moving frame improves the results significantly; and 3) proper filtering in space and time can reduce errors in retrieved wind fields. Two independent cases are studied to test the robustness of the scheme.

## Abstract

Green’s eddy diffusive transfer representation is used to parameterize the meridional eddy heat flux. The structural function obtained by Branscome for the diagonal component *K*
_{yy} in the tensor of the transfer coefficients is adopted. A least squares method that uses the observed data of eddy heat flux is proposed to evaluate the magnitude of *K*
_{yy} and the structure of the nondiagonal component *K*
_{yz} in the transfer coefficient tensor. The optimum motion characteristic at the steering level is used as a constraint for the relationship between *K*
_{yy} and *K*
_{yz}. The obtained magnitude of *K*
_{yy} is two to three times larger than that of the Branscome’s, which is obtained in a linear analysis with the assumption of *K*
_{yz} = 0.

Green’s vertically integrated expression for the meridional eddy momentum flux is used to test the coefficients obtained in the eddy heat flux. In this parameterization, the eddy momentum flux is related to the eddy fluxes of two conserved quantities: potential vorticity and potential temperature. The transfer coefficient is taken to be the sum of that obtained in the parameterization of eddy heat flux, plus a correction term suggested by Stone and Yao, which ensures the global net eddy momentum transport to be zero. What makes the present method attractive is that, even though only the data of eddy heat flux are used to evaluate the magnitude of the transfer coefficients, the obtained magnitude of the eddy momentum flux is in good agreement with observations. For the annual mean calculation, the obtained peak values of eddy momentum flux are 94% of the observation for the Northern Hemisphere and 101% for the Southern Hemisphere. This result significantly improves the result of Stone and Yao, who obtained 34% for the Northern Hemisphere and 16% for the Southern Hemisphere in a similar calculation, but in which *K*
_{yz} = 0 was assumed.

## Abstract

Green’s eddy diffusive transfer representation is used to parameterize the meridional eddy heat flux. The structural function obtained by Branscome for the diagonal component *K*
_{yy} in the tensor of the transfer coefficients is adopted. A least squares method that uses the observed data of eddy heat flux is proposed to evaluate the magnitude of *K*
_{yy} and the structure of the nondiagonal component *K*
_{yz} in the transfer coefficient tensor. The optimum motion characteristic at the steering level is used as a constraint for the relationship between *K*
_{yy} and *K*
_{yz}. The obtained magnitude of *K*
_{yy} is two to three times larger than that of the Branscome’s, which is obtained in a linear analysis with the assumption of *K*
_{yz} = 0.

Green’s vertically integrated expression for the meridional eddy momentum flux is used to test the coefficients obtained in the eddy heat flux. In this parameterization, the eddy momentum flux is related to the eddy fluxes of two conserved quantities: potential vorticity and potential temperature. The transfer coefficient is taken to be the sum of that obtained in the parameterization of eddy heat flux, plus a correction term suggested by Stone and Yao, which ensures the global net eddy momentum transport to be zero. What makes the present method attractive is that, even though only the data of eddy heat flux are used to evaluate the magnitude of the transfer coefficients, the obtained magnitude of the eddy momentum flux is in good agreement with observations. For the annual mean calculation, the obtained peak values of eddy momentum flux are 94% of the observation for the Northern Hemisphere and 101% for the Southern Hemisphere. This result significantly improves the result of Stone and Yao, who obtained 34% for the Northern Hemisphere and 16% for the Southern Hemisphere in a similar calculation, but in which *K*
_{yz} = 0 was assumed.

## Abstract

We examine the fidelity of one-dimensional spectra of vertical velocity derived from multiple-Doppler radar observations. For analytical tractability, we use the following assumptions: 1) the turbulent atmosphere has infinite horizontal dimensions and a vanishing vertical velocity at *x*
_{3} = *H* (due to elevated inversion or a tropopause) and at the surface; 2) the turbulence is isotropic and has a Kolmogorov spectrum; 3) the combined effect of pulse-volume averaging and objective analysis is to replace point measurements by a volume integral; and 4) for the purpose of estimating the horizontal velocities, multiple non-orthogonal radars may be replaced by two equivalent orthogonal radars. The findings are as follows: a) For equal-resolution radars, the relative response decreases monotonically with wavenumber. Above the cutoff wave-number, the relative response is close to zero, while below ∼0.1 of the cutoff wavenumber an asymptotic value less than unity is obtained. This asymptotic value is found to be a function of the vertical resolution. b) Using unequal resolution to resolve some small-scale features should be pursued with caution. Depending on the vertical resolution, the wavenumber and the direction of the line along which the spectrum is taken, some useful small-scale information can be retained, but for some parameter values, unequal resolution can lead to erroneous conclusions. c) Theory and experiment indicate that for flows of finite depth, the attenuation can depend not on the dimensional vertical resolution Δ*z* but on the number of vertical levels that can be resolved by the radars, i.e., on the non-dimensional vertical resolution.

## Abstract

We examine the fidelity of one-dimensional spectra of vertical velocity derived from multiple-Doppler radar observations. For analytical tractability, we use the following assumptions: 1) the turbulent atmosphere has infinite horizontal dimensions and a vanishing vertical velocity at *x*
_{3} = *H* (due to elevated inversion or a tropopause) and at the surface; 2) the turbulence is isotropic and has a Kolmogorov spectrum; 3) the combined effect of pulse-volume averaging and objective analysis is to replace point measurements by a volume integral; and 4) for the purpose of estimating the horizontal velocities, multiple non-orthogonal radars may be replaced by two equivalent orthogonal radars. The findings are as follows: a) For equal-resolution radars, the relative response decreases monotonically with wavenumber. Above the cutoff wave-number, the relative response is close to zero, while below ∼0.1 of the cutoff wavenumber an asymptotic value less than unity is obtained. This asymptotic value is found to be a function of the vertical resolution. b) Using unequal resolution to resolve some small-scale features should be pursued with caution. Depending on the vertical resolution, the wavenumber and the direction of the line along which the spectrum is taken, some useful small-scale information can be retained, but for some parameter values, unequal resolution can lead to erroneous conclusions. c) Theory and experiment indicate that for flows of finite depth, the attenuation can depend not on the dimensional vertical resolution Δ*z* but on the number of vertical levels that can be resolved by the radars, i.e., on the non-dimensional vertical resolution.