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Abstract
It is shown that calculations of the energy conversion from available potential energy to kinetic energy based on vertical velocities obtained by the so-called adiabatic method may lead to erroneous results. An analysis of the method shows that it measures the difference between the energy conversion from available potential energy to kinetic energy, and the generation of available potential by diabatic heating. This result holds for both zonal and eddy avai1able potential energy.
The main conclusion is tested using numerical results from several energy conversion and energy generation studies.
The adiabatic method is compared with other methods to estimate vertical velocities.
Abstract
It is shown that calculations of the energy conversion from available potential energy to kinetic energy based on vertical velocities obtained by the so-called adiabatic method may lead to erroneous results. An analysis of the method shows that it measures the difference between the energy conversion from available potential energy to kinetic energy, and the generation of available potential by diabatic heating. This result holds for both zonal and eddy avai1able potential energy.
The main conclusion is tested using numerical results from several energy conversion and energy generation studies.
The adiabatic method is compared with other methods to estimate vertical velocities.
Meteorology and the Oceans
Lecture presented on the occasion of the award of the Buys Ballot Medal to Dr. A. C. Wiin-Nielsen The Hague, Netherlands, 13 May 1982
Abstract
The mean meridional circulation calculated earlier from observed values of transports of momentum and sensible heat by solving the zonally averaged form of the quasi-geostrophic omega equation is used to investigate the influence of this secondary flow on the zonally averaged values of the wind and temperature in the atmosphere. The contributions from the horizontal transport processes and the mean meridional circulations are computed separately in order to estimate their relative importance. It is found that the mean meridional circulation counterbalances the horizontal transport of momentum in the upper troposphere, while the two effects work in the same direction in the lower part of the atmosphere. With respect to changes in the zonally averaged temperature field, it is found that the effect of the mean meridional circulation opposes the effect of the horizontal transport of sensible heat almost everywhere.
The recent results of calculations of the mean meridional circulation are also used to discuss the role of zonal heating and friction in quasi-geostrophic models.
Abstract
The mean meridional circulation calculated earlier from observed values of transports of momentum and sensible heat by solving the zonally averaged form of the quasi-geostrophic omega equation is used to investigate the influence of this secondary flow on the zonally averaged values of the wind and temperature in the atmosphere. The contributions from the horizontal transport processes and the mean meridional circulations are computed separately in order to estimate their relative importance. It is found that the mean meridional circulation counterbalances the horizontal transport of momentum in the upper troposphere, while the two effects work in the same direction in the lower part of the atmosphere. With respect to changes in the zonally averaged temperature field, it is found that the effect of the mean meridional circulation opposes the effect of the horizontal transport of sensible heat almost everywhere.
The recent results of calculations of the mean meridional circulation are also used to discuss the role of zonal heating and friction in quasi-geostrophic models.
Abstract
The results of extended integrations of a two-level, quasi-geostrophic model with Newtonian heating and dissipation in terms of surface friction, internal friction, and lateral diffusion are described. The major emphasis is on an analysis of the integrations in wavenumber space, including the calculations of spectra for available potential energy, kinetic energy, enstrophy, energy generations, conversions and dissipation, as well as the nonlinear cascades of the first three quantities.
It is found that the fluxes of available potential energy and kinetic energy through the wavenumber domain are very small above planetary wavenumber n = 10, while the enstrophy flux is large and positive for 6 ≤ n ≤ 10, but decreases rapidly for n > 10. The available potential energy, the kinetic energy and the enstrophy as a function of wavenumber vary approximately as n −5, n −3 and n −1, for 10 ≤ n ≤ 20.
Dimensional considerations based on a balance between the convergence of the enstrophy flux and the dissipation of enstrophy in wavenumber space is used to describe the experimental spectra for n > 10. In this analysis it is assumed that the flux of enstrophy is proportional to the product of the wavenumber and the enstrophy divided by a time scale which is related to the flux of enstrophy coming from the baroclinically active region in the wavenumber domain. Theory and experiment are compared with generally good agreement.
Abstract
The results of extended integrations of a two-level, quasi-geostrophic model with Newtonian heating and dissipation in terms of surface friction, internal friction, and lateral diffusion are described. The major emphasis is on an analysis of the integrations in wavenumber space, including the calculations of spectra for available potential energy, kinetic energy, enstrophy, energy generations, conversions and dissipation, as well as the nonlinear cascades of the first three quantities.
It is found that the fluxes of available potential energy and kinetic energy through the wavenumber domain are very small above planetary wavenumber n = 10, while the enstrophy flux is large and positive for 6 ≤ n ≤ 10, but decreases rapidly for n > 10. The available potential energy, the kinetic energy and the enstrophy as a function of wavenumber vary approximately as n −5, n −3 and n −1, for 10 ≤ n ≤ 20.
Dimensional considerations based on a balance between the convergence of the enstrophy flux and the dissipation of enstrophy in wavenumber space is used to describe the experimental spectra for n > 10. In this analysis it is assumed that the flux of enstrophy is proportional to the product of the wavenumber and the enstrophy divided by a time scale which is related to the flux of enstrophy coming from the baroclinically active region in the wavenumber domain. Theory and experiment are compared with generally good agreement.
Abstract
The geostrophic adjustment problem is considered for the case of a homogeneous fluid in a rotating cylindrical container. The formal solution for an arbitrary initial disturbance in the axisymmetrical case is obtained in terms of a series expansion in Bessel functions of zero order. The solution shows that the motion will consist of a time-independent part in geostrophic balance and a time-dependent part which is oscillatory. The general properties of the solution including the energeties are investigated.
The degree to which the process is simulated by various finite-difference methods for the time derivatives is investigated using the leap-frog, a semi-implicit scheme, and a scheme which combines forward and backward time differences (mixed scheme). It is found that all schemes are acceptable provided the time step is sufficiently small, but in general the simulation of the process by the leap-frog and the mixed scheme is more realistic.
The present conclusions are of importance in any scheme employed for the purposes of assimilation of meteorological data. The analysis can be expanded to a more general case than the axisymmetrical one.
Abstract
The geostrophic adjustment problem is considered for the case of a homogeneous fluid in a rotating cylindrical container. The formal solution for an arbitrary initial disturbance in the axisymmetrical case is obtained in terms of a series expansion in Bessel functions of zero order. The solution shows that the motion will consist of a time-independent part in geostrophic balance and a time-dependent part which is oscillatory. The general properties of the solution including the energeties are investigated.
The degree to which the process is simulated by various finite-difference methods for the time derivatives is investigated using the leap-frog, a semi-implicit scheme, and a scheme which combines forward and backward time differences (mixed scheme). It is found that all schemes are acceptable provided the time step is sufficiently small, but in general the simulation of the process by the leap-frog and the mixed scheme is more realistic.
The present conclusions are of importance in any scheme employed for the purposes of assimilation of meteorological data. The analysis can be expanded to a more general case than the axisymmetrical one.
Abstract
The middle-latitude standing wave problem is investigated by means of a quasi-geostrophic, linear, steady-state model in which the zonal current is perturbed by the lower boundary topography and by a distribution of heat sources and sinks. All the perturbations are assumed to have a single meridional wavelength and the dissipation is considered to take place in the surface boundary layer using, as a first approach, a horizontally uniform drag coefficient.
After investigating some basic properties of the model atmosphere, some computations are made to determine its response to the combined forcing by topography and by diabatic heating for January 1962. The resulting perturbations are found to be in rather good agreement with the observed standing waves. The results also indicate that the standing waves forced by the topography are in about the same position as those forced by the diabatic heating and that the former have somewhat larger amplitudes than the latter.
The effect of allowing the drag coefficient to have one constant value over the continents and a smaller constant value over the oceans is examined and found to be quite important when the ratio of the two values is 6, but small (yet such as to bring the computed and observed eddies into closer agreement than in the case of a uniform drag coefficient) for a ratio of 2.
Abstract
The middle-latitude standing wave problem is investigated by means of a quasi-geostrophic, linear, steady-state model in which the zonal current is perturbed by the lower boundary topography and by a distribution of heat sources and sinks. All the perturbations are assumed to have a single meridional wavelength and the dissipation is considered to take place in the surface boundary layer using, as a first approach, a horizontally uniform drag coefficient.
After investigating some basic properties of the model atmosphere, some computations are made to determine its response to the combined forcing by topography and by diabatic heating for January 1962. The resulting perturbations are found to be in rather good agreement with the observed standing waves. The results also indicate that the standing waves forced by the topography are in about the same position as those forced by the diabatic heating and that the former have somewhat larger amplitudes than the latter.
The effect of allowing the drag coefficient to have one constant value over the continents and a smaller constant value over the oceans is examined and found to be quite important when the ratio of the two values is 6, but small (yet such as to bring the computed and observed eddies into closer agreement than in the case of a uniform drag coefficient) for a ratio of 2.
Abstract
The meridional transport of quasi-geostrophic potential vorticity is calculated from atmospheric data from 1 yr. The calculation is based on available data for the transports of relative momentum and sensible heat from eight standard pressure levels. The results show that the transport of potential vorticity is negative (southward) in the major part of the troposphere above the lowest layer of the atmosphere. A small region of positive (northward) transport is found in connection with the subtropical jet stream in summer.
The computed transports of potential vorticity are used in a calculation of the mean annual heat sources in the atmosphere from a steady state quasi-geostrophic model. The results show that the atmosphere is heated south of 50°N and cooled north of this latitude and that the major heat source and heat sink are found around 70 cb. The calculation compares favorably with the determination of the heat sources from heat budget calculation.
It is shown that the transport of potential vorticity is along the gradient of the potential vorticity in the troposphere above approximately 80 cb. Exchange coefficients for the transport of potential vorticity are computed as a function of latitude and pressure for the annual mean values. In addition, exchange coefficients for the transport of sensible heat are obtained, and it is shown that these coefficients are positive in the troposphere below 20 cb.
Abstract
The meridional transport of quasi-geostrophic potential vorticity is calculated from atmospheric data from 1 yr. The calculation is based on available data for the transports of relative momentum and sensible heat from eight standard pressure levels. The results show that the transport of potential vorticity is negative (southward) in the major part of the troposphere above the lowest layer of the atmosphere. A small region of positive (northward) transport is found in connection with the subtropical jet stream in summer.
The computed transports of potential vorticity are used in a calculation of the mean annual heat sources in the atmosphere from a steady state quasi-geostrophic model. The results show that the atmosphere is heated south of 50°N and cooled north of this latitude and that the major heat source and heat sink are found around 70 cb. The calculation compares favorably with the determination of the heat sources from heat budget calculation.
It is shown that the transport of potential vorticity is along the gradient of the potential vorticity in the troposphere above approximately 80 cb. Exchange coefficients for the transport of potential vorticity are computed as a function of latitude and pressure for the annual mean values. In addition, exchange coefficients for the transport of sensible heat are obtained, and it is shown that these coefficients are positive in the troposphere below 20 cb.
Abstract
A two level quasi-geostrophic model for zonally averaged conditions has been integrated for a period of a few years. The model is forced by Newtonian heating and has internal and surface friction. The interaction between the zonal flow and the eddies is simulated through the use of exchange coefficients for the transports of quasi-geostrophic potential vorticity and sensible heat.
The results of the integrations show that the model predicts a qualitatively correct annual variation of the zonal winds and the zonal temperature, although the predicted annual cycle has a too large amplitude compared with observations. The times of the maximum amounts of available potential and kinetic energy are well predicted as well as the typical time lag between the two quantities. The same statement holds for the generation of zonal available potential energy and the dissipation of zonal kinetic energy. The energy diagram obtained as an average for 1 yr of integration compares well with the corresponding diagram based on observations.
The major weakness of the model (i.e., the large annual variation of most quantities) is probably related to the simplicity of the thermal forcing.
Abstract
A two level quasi-geostrophic model for zonally averaged conditions has been integrated for a period of a few years. The model is forced by Newtonian heating and has internal and surface friction. The interaction between the zonal flow and the eddies is simulated through the use of exchange coefficients for the transports of quasi-geostrophic potential vorticity and sensible heat.
The results of the integrations show that the model predicts a qualitatively correct annual variation of the zonal winds and the zonal temperature, although the predicted annual cycle has a too large amplitude compared with observations. The times of the maximum amounts of available potential and kinetic energy are well predicted as well as the typical time lag between the two quantities. The same statement holds for the generation of zonal available potential energy and the dissipation of zonal kinetic energy. The energy diagram obtained as an average for 1 yr of integration compares well with the corresponding diagram based on observations.
The major weakness of the model (i.e., the large annual variation of most quantities) is probably related to the simplicity of the thermal forcing.
Abstract
The energy conversion between the vertical shear flow and the vertical mean flow has been computed using atmospheric data from the isobaric surfaces: 850, 700,500, 300, and 200 mb. In comparison with earlier calculations based on a smaller vertical resolution (2 levels) and a smaller sample, it is found that the new calculations give larger numerical values in better agreement with the results of numerical experiments concerning the general circulation of the atmosphere. The energy transformation has been computed in the wave number regime, and it is found that the medium-scale waves are responsible for the major portion of the transformation.
The amounts of energy in the baroclinic component (the vertical shear flow) and the barotropic component (the vertical mean flow) have been computed as a function of wave number. It is found that the kinetic energy in the barotropic component is about 2.6 times the kinetic energy in the baroclinic component. The partitioning of the kinetic energy between the zonal flow and the eddies is such that the eddies contain more energy than the zonal flow. This result applies for the vertical shear flow as well as the vertical mean flow and is in contrast to the results obtained from numerical experiments regarding the general circulation.
The present computations include only the energy calculations which would be present in a quasi-non-divergent model. Later calculations will provide estimates of the remaining term of the energy conversion.
Abstract
The energy conversion between the vertical shear flow and the vertical mean flow has been computed using atmospheric data from the isobaric surfaces: 850, 700,500, 300, and 200 mb. In comparison with earlier calculations based on a smaller vertical resolution (2 levels) and a smaller sample, it is found that the new calculations give larger numerical values in better agreement with the results of numerical experiments concerning the general circulation of the atmosphere. The energy transformation has been computed in the wave number regime, and it is found that the medium-scale waves are responsible for the major portion of the transformation.
The amounts of energy in the baroclinic component (the vertical shear flow) and the barotropic component (the vertical mean flow) have been computed as a function of wave number. It is found that the kinetic energy in the barotropic component is about 2.6 times the kinetic energy in the baroclinic component. The partitioning of the kinetic energy between the zonal flow and the eddies is such that the eddies contain more energy than the zonal flow. This result applies for the vertical shear flow as well as the vertical mean flow and is in contrast to the results obtained from numerical experiments regarding the general circulation.
The present computations include only the energy calculations which would be present in a quasi-non-divergent model. Later calculations will provide estimates of the remaining term of the energy conversion.
Abstract
The contribution from the divergent part of the horizontal wind to the energy conversion between the vertical shear flow and the vertical mean flow has been computed using atmospheric data from the isobaric surfaces: 850, 700, 500, 300, and 200 mb. The new calculations supplement earlier computations giving the energy conversion based on an assumption that the horizontal winds are non-divergent.
It is found that the contribution from the divergent part of the horizontal wind normally is very small compared with the contribution from the non-divergent part. The former energy conversion is as a matter of fact generally not significantly different from zero.
The abnormal winter 1962–63 has been investigated separately. It is found that energy conversion by the divergent wind component during this period was much larger and constituted a larger fraction of the total conversion than during any other period.
Abstract
The contribution from the divergent part of the horizontal wind to the energy conversion between the vertical shear flow and the vertical mean flow has been computed using atmospheric data from the isobaric surfaces: 850, 700, 500, 300, and 200 mb. The new calculations supplement earlier computations giving the energy conversion based on an assumption that the horizontal winds are non-divergent.
It is found that the contribution from the divergent part of the horizontal wind normally is very small compared with the contribution from the non-divergent part. The former energy conversion is as a matter of fact generally not significantly different from zero.
The abnormal winter 1962–63 has been investigated separately. It is found that energy conversion by the divergent wind component during this period was much larger and constituted a larger fraction of the total conversion than during any other period.
