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Abstract
The zonally averaged conservation equations for water, linear zonal momentum, and potential heat (gz+c_{p}T) are written in a form analogous to the mass continuity equation. This is possible when atmospheric storage terms are negligible which is generally the case during the solstice seasons. It follows that the fluxes of these properties can be represented by Stokes streamfunctions. Patterns of streamfunctions in the verticalmeridional plane for mass, water (all 3 phases combined), momentum, and heat have been prepared from the “Atmospheric Circulation Statistics” of Oort and Rasmusson (1971). They are shown for the seasons December–January–February and June–July–August, and for the area between 10S75N. The following details are of interest:

The presented data are not new in any sense. Only the mode of presentation is new.

Contrary to the total mass transport, which is almost entirely conservative, all other transports have sources and sinks. They are treated as vertical flux divergences and thus are amenable to the streamfunction concept.

The streamfunction pattern can be considerably modified by linear combination with the mass continuity equation, characterized by a scale function: This function is zero for water transport, a ^{2}Ω cos^{2}ϕ for momentum transport, and gz_{0} +c_{p}T _{0} for heat transport (z _{0}, T _{0} averages over total atmosphere). This choice minimizes the backandforth transport of properties by the cell circulation.

Boundary conditions are that the upper surface of the atmosphere be a streamline for mass, water, and momentum transport. For potential heat, the value at the upper surface at a certain latitude is the net radiation flux across this surface, integrated between this latitude and the pole.

The streamlines represent the total flux of the respective property in whatever form. For instance, vertical fluxes of water comprise mean and eddy components of all scales as well as net contributions of solid and liquid water flux.

The streamlines of transports with sources and sinks begin and end at the earth's surface (water, momentum, and heat) or additionally at the upper surface of the atmosphere (heat). There are no closed isolines.
The mixed character of the various fluxes is qualitatively described. Fluxes of different properties cross each other or go in opposite directions. Further, fluxes of the same property on different scales may go in opposite directions, particularly in the vertical. The total horizontal flux divergences are compared with some independent flux estimates at the earth's surface. Although there are still significant imbalances, the general agreement is fair.
Abstract
The zonally averaged conservation equations for water, linear zonal momentum, and potential heat (gz+c_{p}T) are written in a form analogous to the mass continuity equation. This is possible when atmospheric storage terms are negligible which is generally the case during the solstice seasons. It follows that the fluxes of these properties can be represented by Stokes streamfunctions. Patterns of streamfunctions in the verticalmeridional plane for mass, water (all 3 phases combined), momentum, and heat have been prepared from the “Atmospheric Circulation Statistics” of Oort and Rasmusson (1971). They are shown for the seasons December–January–February and June–July–August, and for the area between 10S75N. The following details are of interest:

The presented data are not new in any sense. Only the mode of presentation is new.

Contrary to the total mass transport, which is almost entirely conservative, all other transports have sources and sinks. They are treated as vertical flux divergences and thus are amenable to the streamfunction concept.

The streamfunction pattern can be considerably modified by linear combination with the mass continuity equation, characterized by a scale function: This function is zero for water transport, a ^{2}Ω cos^{2}ϕ for momentum transport, and gz_{0} +c_{p}T _{0} for heat transport (z _{0}, T _{0} averages over total atmosphere). This choice minimizes the backandforth transport of properties by the cell circulation.

Boundary conditions are that the upper surface of the atmosphere be a streamline for mass, water, and momentum transport. For potential heat, the value at the upper surface at a certain latitude is the net radiation flux across this surface, integrated between this latitude and the pole.

The streamlines represent the total flux of the respective property in whatever form. For instance, vertical fluxes of water comprise mean and eddy components of all scales as well as net contributions of solid and liquid water flux.

The streamlines of transports with sources and sinks begin and end at the earth's surface (water, momentum, and heat) or additionally at the upper surface of the atmosphere (heat). There are no closed isolines.
The mixed character of the various fluxes is qualitatively described. Fluxes of different properties cross each other or go in opposite directions. Further, fluxes of the same property on different scales may go in opposite directions, particularly in the vertical. The total horizontal flux divergences are compared with some independent flux estimates at the earth's surface. Although there are still significant imbalances, the general agreement is fair.
Abstract
From surface wind estimates published in the Dutch Atlas monthly charts of surface wind vergence over the Indian Ocean down to 50S were computed. Since the original data were smoothed by a lowpass filter, the charts exhibit only regional and largescale features. The vergence distributions are not zonally symmetric; rather, they show a celllike structure.
The vergence patterns are discussed in terms of latitudinaltime sections, both for the Arabian Sea and the Bay of Bengal longitude range, and compared with similar plots of the precipitation frequency. North of 10–20S there seem to exist three different circulation regimes, separated by sharply defined transition periods, a characteristic of the Indian monsoon climate. A simple description of this threefold monsoonal rhythm is given in terms of the first and second harmonies of the annual march of temperature.
Abstract
From surface wind estimates published in the Dutch Atlas monthly charts of surface wind vergence over the Indian Ocean down to 50S were computed. Since the original data were smoothed by a lowpass filter, the charts exhibit only regional and largescale features. The vergence distributions are not zonally symmetric; rather, they show a celllike structure.
The vergence patterns are discussed in terms of latitudinaltime sections, both for the Arabian Sea and the Bay of Bengal longitude range, and compared with similar plots of the precipitation frequency. North of 10–20S there seem to exist three different circulation regimes, separated by sharply defined transition periods, a characteristic of the Indian monsoon climate. A simple description of this threefold monsoonal rhythm is given in terms of the first and second harmonies of the annual march of temperature.
Abstract
We consider global climate models based on zonally averaged balance relations. Inherent boundary conditions require the meridional fluxes of nonnegative properties (temperature, humidity, energy, etc.), as well as the flux of zonal momentum, to vanish at both poles. On the other hand, the meridional divergence of these fluxes does not vanish at either pole. An important exception from this general nonzero polar divergence condition of meridional fluxes is the transport of zonal momentum; the meridional divergence of zonal momentum flux vanishes at the pole because there is neither zonal surface stress nor horizontal wind. These conditions are derived from the balance equations for energy and momentum. Furthermore, they are tested with observed flux data for specific humidity and zonal wind. The closure problem in such models is often overcome by a diffusive parameterization of the fluxes in terms of meridional gradients. It is shown that, due to the above conditions, the exchange coefficient for the energy transport may not vanish at the poles. This has implications for semiempirical models designed to test climate's stability and transitivity.
Abstract
We consider global climate models based on zonally averaged balance relations. Inherent boundary conditions require the meridional fluxes of nonnegative properties (temperature, humidity, energy, etc.), as well as the flux of zonal momentum, to vanish at both poles. On the other hand, the meridional divergence of these fluxes does not vanish at either pole. An important exception from this general nonzero polar divergence condition of meridional fluxes is the transport of zonal momentum; the meridional divergence of zonal momentum flux vanishes at the pole because there is neither zonal surface stress nor horizontal wind. These conditions are derived from the balance equations for energy and momentum. Furthermore, they are tested with observed flux data for specific humidity and zonal wind. The closure problem in such models is often overcome by a diffusive parameterization of the fluxes in terms of meridional gradients. It is shown that, due to the above conditions, the exchange coefficient for the energy transport may not vanish at the poles. This has implications for semiempirical models designed to test climate's stability and transitivity.
Abstract
A nonlinear, timedependent, baroclinic model is developed for a zonally uniform, tropical, twolayer ocean on a northsouth vertical section. The lower layer is infinitely deep, at rest, and at constant temperature. The dynamics of the wellmixed surface layer are described in terms of the components of horizontal mass transport, the specific mass, and the specific enthalpy. The forcing functions of the model are the zonal wind stress, the vertical entrainment of cold water from the lower layer into the surface layer, and the surface thermal energy input. The concept of entrainment forcing is based on the approach of Kraus and Turner for parameterizing the vertical motion of the seasonal thermocline.
Since zonal gradients of all quantities are neglected, the model applies only to the ocean's interior. This is rationalized by oceanographical observations. In particular, the castwest pressure gradient term is one order of magnitude smaller than the wind stress; it may be considered as an additional forcing function and, as such, absorbed in the zonal wind stress. Scale analysis reveals two time scales inherent in the model: a short scale of 0.2 day governing the mass transport equations, and a long scale of several years governing the conservation equations for mass and enthalpy. Shortterm climatic fluctuations may be controlled by the latter scale.
Solutions for the steady state with Rossby number zero are presented. For wind stress and thermal energy input, simple analytic functions similar in shape to observed patterns are used. For the entrainment function three different possible distributions are investigated, all of which have an equatorial maximum attributed to strong vertical mixing in the equatorial undercurrent region. The principle responses of the model are: 1) a meridional pattern of zonal mass transport exhibiting the main observed features, particularly an equatorial countercurrent; 2) a thickness of the mixed layer similar to the observations; and 3) a surface temperature profile with an equatorial minimum. The tendency of this model to develop an equatorial countercurrent is caused by the entrainment forcing. It is shown that entrainment and energy balance are not entirely independent of one another.
Abstract
A nonlinear, timedependent, baroclinic model is developed for a zonally uniform, tropical, twolayer ocean on a northsouth vertical section. The lower layer is infinitely deep, at rest, and at constant temperature. The dynamics of the wellmixed surface layer are described in terms of the components of horizontal mass transport, the specific mass, and the specific enthalpy. The forcing functions of the model are the zonal wind stress, the vertical entrainment of cold water from the lower layer into the surface layer, and the surface thermal energy input. The concept of entrainment forcing is based on the approach of Kraus and Turner for parameterizing the vertical motion of the seasonal thermocline.
Since zonal gradients of all quantities are neglected, the model applies only to the ocean's interior. This is rationalized by oceanographical observations. In particular, the castwest pressure gradient term is one order of magnitude smaller than the wind stress; it may be considered as an additional forcing function and, as such, absorbed in the zonal wind stress. Scale analysis reveals two time scales inherent in the model: a short scale of 0.2 day governing the mass transport equations, and a long scale of several years governing the conservation equations for mass and enthalpy. Shortterm climatic fluctuations may be controlled by the latter scale.
Solutions for the steady state with Rossby number zero are presented. For wind stress and thermal energy input, simple analytic functions similar in shape to observed patterns are used. For the entrainment function three different possible distributions are investigated, all of which have an equatorial maximum attributed to strong vertical mixing in the equatorial undercurrent region. The principle responses of the model are: 1) a meridional pattern of zonal mass transport exhibiting the main observed features, particularly an equatorial countercurrent; 2) a thickness of the mixed layer similar to the observations; and 3) a surface temperature profile with an equatorial minimum. The tendency of this model to develop an equatorial countercurrent is caused by the entrainment forcing. It is shown that entrainment and energy balance are not entirely independent of one another.
Abstract
The surface wind stress curl is the forcing function in the equations of vertically integrated water transport of winddriven ocean currents. Hence, it has become a basic quantity in theoretical oceanography. As the time dependence of all important surface quantities in the Indian Ocean is stronger than in other oceans, it is valuable to look particularly at the time variation in this region. This study presents monthly charts of the wind stress curl at the surface of the Indian Ocean from its land boundaries up to 50° S. and from 20° E. to 116° E.
Basic data were the monthly surface maps of the Koninklijk Nederlands Meteorologisch Instituut, derived from ship observations and given as 2° square means of the surface wind. The processing of the data is described in detail. In particular, smallscale fluctuations are objectively filtered out.
While earlier compilations are usually on a coarser grid (seasonal and 5° square averages), the present data have a refined time and space resolution. Therefore, they allow one to study more detailed structures. In particular, the charts show that the curl pattern in the Indian Ocean is not independent of longitude.
Abstract
The surface wind stress curl is the forcing function in the equations of vertically integrated water transport of winddriven ocean currents. Hence, it has become a basic quantity in theoretical oceanography. As the time dependence of all important surface quantities in the Indian Ocean is stronger than in other oceans, it is valuable to look particularly at the time variation in this region. This study presents monthly charts of the wind stress curl at the surface of the Indian Ocean from its land boundaries up to 50° S. and from 20° E. to 116° E.
Basic data were the monthly surface maps of the Koninklijk Nederlands Meteorologisch Instituut, derived from ship observations and given as 2° square means of the surface wind. The processing of the data is described in detail. In particular, smallscale fluctuations are objectively filtered out.
While earlier compilations are usually on a coarser grid (seasonal and 5° square averages), the present data have a refined time and space resolution. Therefore, they allow one to study more detailed structures. In particular, the charts show that the curl pattern in the Indian Ocean is not independent of longitude.
Abstract
The diabatic heating of the atmosphere can be very closely represented by the convergence of a vertical energy flux. It consists of two components: the flux of net radiation and the flux of precipitation. The latter comprises the vertical flux of water in condensed form (rain, snow, ice). The concept of precipitation flux is investigated employing the zonal mean equation of potential heat. Input data are radiation flux from a model, adjusted at the top and bottom of the atmosphere with observed data; horizontal advective heat flux convergence and heat storage with the data of the MIT Library; and vertical subsynoptic eddy flux of sensible heat (a small quantity) from a parameterization. Output is the precipitation flux in the free atmosphere. Time scale is 1 month, space domain is the zonal mean Northern Hemisphere.
The precipitation flux is downward everywhere. It is maximum in the tropics. Comparison of the flux across the 1000 mb level with the observed surface precipitation shows satisfactory agreement. The balance in the potential heat equation is largely between radiation and precipitation; thus the atmosphere can be characterized by an approximate radiativeprecipitative equilibrium. The accuracy of the method (±10 W m^{−2}) depends critically on the validity of the radiation data.
Abstract
The diabatic heating of the atmosphere can be very closely represented by the convergence of a vertical energy flux. It consists of two components: the flux of net radiation and the flux of precipitation. The latter comprises the vertical flux of water in condensed form (rain, snow, ice). The concept of precipitation flux is investigated employing the zonal mean equation of potential heat. Input data are radiation flux from a model, adjusted at the top and bottom of the atmosphere with observed data; horizontal advective heat flux convergence and heat storage with the data of the MIT Library; and vertical subsynoptic eddy flux of sensible heat (a small quantity) from a parameterization. Output is the precipitation flux in the free atmosphere. Time scale is 1 month, space domain is the zonal mean Northern Hemisphere.
The precipitation flux is downward everywhere. It is maximum in the tropics. Comparison of the flux across the 1000 mb level with the observed surface precipitation shows satisfactory agreement. The balance in the potential heat equation is largely between radiation and precipitation; thus the atmosphere can be characterized by an approximate radiativeprecipitative equilibrium. The accuracy of the method (±10 W m^{−2}) depends critically on the validity of the radiation data.
Abstract
The zonal mean flux vector of atmospheric heat transports is very closely nondivergent in the verticalmeridional plane. This is demonstrated for potential (c_{p}T + gz) and latent heat. Thus the heat flux vector fields can be represented by streamfunctions. The top of the atmosphere is a streamline for latent heat. For potential heat, the radiation flux across the top determines the upper boundary condition.
The conservation equations are invariant with respect to arbitrarily choosing a constant reference heat but the streamfunctions are not. The impact on the streamfunctions of shifting the reference heat is equivalent to subtracting the mass transport, scaled with that constant, from the heat flux. To remove this ambiguity it is postulated that the curt of the heat flux vector in the verticalmeridional plane be minimized in the leastsquares sense. This principle of minimum mean enstrophy is rationalized by analogy to electrodynamics. It yields a formula for the reference heat in terms of hemispheric integrals of heat and mass flux curl. The formula is applied to the circulation statistics of the MITLibrary. The reference constants turn out to be numerically identical to the observed hemispheric annual mean of the respective heat form (∼324 J g^{−1} for potential heat and ∼6 J g^{−1}, corresponding to 2.6 g kg^{−1}, for latent heat).
Streamfunctions reduced in this way are presented for the seasons. The potential heat circulation is highly variable. It changes sign from summer to winter over the entire northern atmosphere. The water circulation is less variable; it changes sign only in the tropics and fluctuates in intensity in the extratropics. It is shown that there is no further ambiguity in the streamfunction concept. The interrelationships between kinetic energy and potential, latent and static heat flux are discussed with the main result that the potential heat circulation is largely governed by radiation and precipitation flux but very little by the kinetic energy flux and not at all by the water vapor, and further that the streamfunction concept is not applicable to the kinetic energy flux. The main virtue of the streamfunctions is that they quantitatively represent the net heat flux on all scales and phases.
Abstract
The zonal mean flux vector of atmospheric heat transports is very closely nondivergent in the verticalmeridional plane. This is demonstrated for potential (c_{p}T + gz) and latent heat. Thus the heat flux vector fields can be represented by streamfunctions. The top of the atmosphere is a streamline for latent heat. For potential heat, the radiation flux across the top determines the upper boundary condition.
The conservation equations are invariant with respect to arbitrarily choosing a constant reference heat but the streamfunctions are not. The impact on the streamfunctions of shifting the reference heat is equivalent to subtracting the mass transport, scaled with that constant, from the heat flux. To remove this ambiguity it is postulated that the curt of the heat flux vector in the verticalmeridional plane be minimized in the leastsquares sense. This principle of minimum mean enstrophy is rationalized by analogy to electrodynamics. It yields a formula for the reference heat in terms of hemispheric integrals of heat and mass flux curl. The formula is applied to the circulation statistics of the MITLibrary. The reference constants turn out to be numerically identical to the observed hemispheric annual mean of the respective heat form (∼324 J g^{−1} for potential heat and ∼6 J g^{−1}, corresponding to 2.6 g kg^{−1}, for latent heat).
Streamfunctions reduced in this way are presented for the seasons. The potential heat circulation is highly variable. It changes sign from summer to winter over the entire northern atmosphere. The water circulation is less variable; it changes sign only in the tropics and fluctuates in intensity in the extratropics. It is shown that there is no further ambiguity in the streamfunction concept. The interrelationships between kinetic energy and potential, latent and static heat flux are discussed with the main result that the potential heat circulation is largely governed by radiation and precipitation flux but very little by the kinetic energy flux and not at all by the water vapor, and further that the streamfunction concept is not applicable to the kinetic energy flux. The main virtue of the streamfunctions is that they quantitatively represent the net heat flux on all scales and phases.
Abstract
The diabatic heating Q is the ultimate driving force of the general circulation and climate. We present seasonal and zonal mean estimates of Q (order of magnitude 10^{−5} K s^{−1}) for the atmosphere from 15°S90°N and from 50–1000 mb. Q comprises radiation, condensation, conduction, dissipation and diffusion; the first two terms are large, the last three are small. We compile Q indirectly by specifying (from the synoptic general circulation statistics of the MIT Library) sensible heat advective and storage terms, including the adiabatic heating, which together balance Q in the First Law of thermodynamics. An important component of the advective terms is subsynopticscale advection. We show that it is not restricted to boundary layer heating but also has convectivescale components of potential significance and seems to be active even in the stratosphere. However, we are not able to specify the total subsynopticscale advection since it is subject to considerable compensation. This causes a systematic error of the order of 10^{−6} K s^{−1} in our synoptic estimates of Q.
The meridional diabatic heating profiles show four latitude belts of different Q climate. The tropics and midlatitudes are characterized by net heating, the subtropics and the polar cap by net cooling. This pattern is visible throughout the year and reflects the net effect of the two governing, and partly balancing, components of Q: condensational heating dominates in the rainbelts, radiational cooling dominates in the dry belts. A new feature in the Q field is persistent strong beating at and above the jet stream level between 30–40°N throughout the year. We speculatively explain this effect with the subsynoptic advective terms.
Abstract
The diabatic heating Q is the ultimate driving force of the general circulation and climate. We present seasonal and zonal mean estimates of Q (order of magnitude 10^{−5} K s^{−1}) for the atmosphere from 15°S90°N and from 50–1000 mb. Q comprises radiation, condensation, conduction, dissipation and diffusion; the first two terms are large, the last three are small. We compile Q indirectly by specifying (from the synoptic general circulation statistics of the MIT Library) sensible heat advective and storage terms, including the adiabatic heating, which together balance Q in the First Law of thermodynamics. An important component of the advective terms is subsynopticscale advection. We show that it is not restricted to boundary layer heating but also has convectivescale components of potential significance and seems to be active even in the stratosphere. However, we are not able to specify the total subsynopticscale advection since it is subject to considerable compensation. This causes a systematic error of the order of 10^{−6} K s^{−1} in our synoptic estimates of Q.
The meridional diabatic heating profiles show four latitude belts of different Q climate. The tropics and midlatitudes are characterized by net heating, the subtropics and the polar cap by net cooling. This pattern is visible throughout the year and reflects the net effect of the two governing, and partly balancing, components of Q: condensational heating dominates in the rainbelts, radiational cooling dominates in the dry belts. A new feature in the Q field is persistent strong beating at and above the jet stream level between 30–40°N throughout the year. We speculatively explain this effect with the subsynoptic advective terms.
Abstract
Snow cover duration is commonly derived from snow depth, snow water equivalent, or satellite data. Snow cover duration has more recently also been inferred from ground temperature data. In this study, a probabilistic snow cover duration (SCD) model is introduced that estimates the conditional probability for snow cover given the daily mean and the diurnal range of ground temperature. For the application of the SCD model, 87 Austrian sites in the Alpine region are investigated in the period of 2000 to 2011. The daily range of ground temperature is identified to represent the primary variable in determining the snow cover duration. In the case of a large dataset, however, the inclusion of the daily mean ground temperature as the second given parameter improves results. Rank correlation coefficients of predicted versus observed snow cover duration are typically between 0.8 and 0.9.
Abstract
Snow cover duration is commonly derived from snow depth, snow water equivalent, or satellite data. Snow cover duration has more recently also been inferred from ground temperature data. In this study, a probabilistic snow cover duration (SCD) model is introduced that estimates the conditional probability for snow cover given the daily mean and the diurnal range of ground temperature. For the application of the SCD model, 87 Austrian sites in the Alpine region are investigated in the period of 2000 to 2011. The daily range of ground temperature is identified to represent the primary variable in determining the snow cover duration. In the case of a large dataset, however, the inclusion of the daily mean ground temperature as the second given parameter improves results. Rank correlation coefficients of predicted versus observed snow cover duration are typically between 0.8 and 0.9.