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- Author or Editor: Tzvi Gal-Chen x
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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
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
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
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 Δz 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 Δz but on the number of vertical levels that can be resolved by the radars, i.e., on the non-dimensional vertical resolution.
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
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 technique developed by Gal-Chen in 1978 is used to derive vertical velocities, buoyancy, and pressure perturbations from dual-Doppler radar observations of the planetary boundary layer (PBL). Several approaches to verification are pursued. They include: (a) scan-to-scan temporal continuity of the derived fields; (b) an objective test to find out how well the derived pressure perturbations balance the dynamical equations; (c) comparison of dual-Doppler derived, horizontally averaged fluxes of heat versus in situ measurements and other data sets; and (d) a noteworthy improvement in the quality of the retrieved pressure when tendencies are included.
Previous studies indicate that in order for the method to be viable the radars have to resolve the PBL with at least ten vertical levels. One such event occurred on 27 September 1978 during project PHOENIX, conducted at the Boulder Atmospheric Observatory (BAO) 300 m tower. An inversion above a shallow boundary layer of height around 800 m was eroded, and the PBL grew to a height of 2.4 km in less than half an hour. During that period, the vertical profiles of potential temperature and pressure variance derived from the two NOAA/Wave Propagation Laboratory X-band (3 cm wavelength) Doppler radars suggest the existence of two inversions. Two inversions are also indicated by the aircraft data.
Some aspects of the derived heat flux profiles, such as negative heat flux at the top of the mixed layer, are classical and constitute further evidence of the plausibility of the results. Some other aspects such as positive vertical gradient of the heat flux profile near the first inversion (where the heat flux is still positive) are not commonly observed. Based on the available data, it is speculated that this latter feature is transient, indicative of the mixing (during the growth of the PBL) of the potentially warmer upper layer with the potentially colder lower layer.
Several closure approximations for three-dimensional PBL models are tested. Nonlinear eddy viscosities are derived from the observed second moments of the Doppler spectrum and are used to estimate the frictional dissipation in a three-dimensional numerical model of the PBL. Except near the ground, the derived temperature and pressure are only slightly sensitive to factor-of-two variation in the value of the eddy viscosity. Furthermore, it is found that adding frictional dissipation does not reduce the imbalance between the horizontal pressure gradient and the horizontal accelerations. Recalling that in a “perfect” three-dimensional model exact balance must prevail, one concludes that this particular subgrid parameterization could be merely a device to prevent excessive accumulation of energy in the smallest resolvable scale of a numerical model.
Abstract
The technique developed by Gal-Chen in 1978 is used to derive vertical velocities, buoyancy, and pressure perturbations from dual-Doppler radar observations of the planetary boundary layer (PBL). Several approaches to verification are pursued. They include: (a) scan-to-scan temporal continuity of the derived fields; (b) an objective test to find out how well the derived pressure perturbations balance the dynamical equations; (c) comparison of dual-Doppler derived, horizontally averaged fluxes of heat versus in situ measurements and other data sets; and (d) a noteworthy improvement in the quality of the retrieved pressure when tendencies are included.
Previous studies indicate that in order for the method to be viable the radars have to resolve the PBL with at least ten vertical levels. One such event occurred on 27 September 1978 during project PHOENIX, conducted at the Boulder Atmospheric Observatory (BAO) 300 m tower. An inversion above a shallow boundary layer of height around 800 m was eroded, and the PBL grew to a height of 2.4 km in less than half an hour. During that period, the vertical profiles of potential temperature and pressure variance derived from the two NOAA/Wave Propagation Laboratory X-band (3 cm wavelength) Doppler radars suggest the existence of two inversions. Two inversions are also indicated by the aircraft data.
Some aspects of the derived heat flux profiles, such as negative heat flux at the top of the mixed layer, are classical and constitute further evidence of the plausibility of the results. Some other aspects such as positive vertical gradient of the heat flux profile near the first inversion (where the heat flux is still positive) are not commonly observed. Based on the available data, it is speculated that this latter feature is transient, indicative of the mixing (during the growth of the PBL) of the potentially warmer upper layer with the potentially colder lower layer.
Several closure approximations for three-dimensional PBL models are tested. Nonlinear eddy viscosities are derived from the observed second moments of the Doppler spectrum and are used to estimate the frictional dissipation in a three-dimensional numerical model of the PBL. Except near the ground, the derived temperature and pressure are only slightly sensitive to factor-of-two variation in the value of the eddy viscosity. Furthermore, it is found that adding frictional dissipation does not reduce the imbalance between the horizontal pressure gradient and the horizontal accelerations. Recalling that in a “perfect” three-dimensional model exact balance must prevail, one concludes that this particular subgrid parameterization could be merely a device to prevent excessive accumulation of energy in the smallest resolvable scale of a numerical model.
Abstract
Thermodynamic analysis indicates, in contradiction to an earlier hypothesis, that a mixture of warm, dry air and cool, moist air on opposite sides of a dryline will have a slightly lower virtual temperature than the average of its components. Such mixing is unlikely, therefore, to contribute directly to dryline convection or maintenance. The correction is due to consideration of the effect of moisture content on specific heat, the neglect of which could lead to significant errors in numerical simulation models.
Abstract
Thermodynamic analysis indicates, in contradiction to an earlier hypothesis, that a mixture of warm, dry air and cool, moist air on opposite sides of a dryline will have a slightly lower virtual temperature than the average of its components. Such mixing is unlikely, therefore, to contribute directly to dryline convection or maintenance. The correction is due to consideration of the effect of moisture content on specific heat, the neglect of which could lead to significant errors in numerical simulation models.
