The results of qualitative laboratory and numerical experiments on two-dimensional non-linear mod... more The results of qualitative laboratory and numerical experiments on two-dimensional non-linear model are described, aiming at an investigation of the structure of the front of bottom gravity current. Non-coincidence of frontal interfaces in density and velocity fields within the bottom gravity current leading (frontal) part is stated on the base of comparative analysis of numerical and laboratory experiments. This fact is experimentally confirmed by field example of marine water inflow into a brackish lagoon. The density gradient along the stream line is shown to be an additional effective criterion for the localization of the frontal zone.
The results of qualitative laboratory and numerical experiments on two-dimensional non-linear mod... more The results of qualitative laboratory and numerical experiments on two-dimensional non-linear model are described, aiming at an investigation of the structure of the front of bottom gravity current. Non-coincidence of frontal interfaces in density and velocity fields within the bottom gravity current leading (frontal) part is stated on the base of comparative analysis of numerical and laboratory experiments. This fact is experimentally confirmed by field example of marine water inflow into a brackish lagoon. The density gradient along the stream line is shown to be an additional effective criterion for the localization of the frontal zone.
Development of a thermal bar in a laboratory flume with an inclined bottom (3.7°–12°) under the c... more Development of a thermal bar in a laboratory flume with an inclined bottom (3.7°–12°) under the conditions of cooling/heating of the water with a temperature close to that of the maximal density is studied. The structure of the temperature field and currents during different stages of the circulation is examined: (i) formation of an along-slope gravity current, (ii) generation of a subsurface jet, and (iii) transformation of one type of the circulation into another at passing the temperature of the maximum density. The “fall” and “spring” types of the thermal bar are shown to be dynamically equivalent: the transport of the near-shore waters to the deepwater part, which is driven by the buoyancy flux rather than by the heat flux across the surface, transforms stage (i) into stage (ii), while the opposite (on-shore) flow is generated in the intermediate layers. A comparison of the results with the field and laboratory data published allows us to suggest that the propagation of the thermal bar front in the “fast” stage can be considered as the development of a convective jet with its velocity U ∼ h 3/4, which is proportional to the growing thickness of the upper layer h affected by the heating/cooling processes
We report the results of laboratory experiments on water heating/cooling, performed in 5 m long w... more We report the results of laboratory experiments on water heating/cooling, performed in 5 m long water channel with a slope. About 63 series of photos were analyzed: for 3 locations, for 3 bottom slopes (3.7, 6.7 and 12 degrees) and for different Ra numbers. It was pointed out that there exist two types of mixing characterizing different circulations in the presence of slope: gravity current and undersurface jet; the thermal bar is the region where one type of mixing is replaced by another; the highest speed and flowrate are at the break point; the flow is three-dimensional.
Development of a thermal bar in a laboratory flume with an inclined bottom (3.7° 12°) under the c... more Development of a thermal bar in a laboratory flume with an inclined bottom (3.7° 12°) under the conditions of cooling/heating of the water with a temperature close to that of the maximal density is studied. The structure of the temperature field and currents during different stages of the circulation is examined: (i) formation of an along-slope gravity current, (ii) generation of a subsurface jet, and (iii) transformation of one type of the circulation into another at passing the temperature of the maximum density. The “fall” and “spring” types of the thermal bar are shown to be dynamically equivalent: the transport of the near-shore waters to the deepwater part, which is driven by the buoyancy flux rather than by the heat flux across the surface, transforms stage (i) into stage (ii), while the opposite (on-shore) flow is generated in the intermediate layers. A comparison of the results with the field and laboratory data published allows us to suggest that the propagation of the thermal bar front in the “fast” stage can be considered as the development of a convective jet with its velocity U ˜ h 3/4, which is proportional to the growing thickness of the upper layer h affected by the heating/cooling processes
The man-made contribution to significant increase in salinity in the Vistula Lagoon (south-easter... more The man-made contribution to significant increase in salinity in the Vistula Lagoon (south-eastern Baltic) during the last century is discussed in this paper: (a) diversion of the main part of the Vistula River discharge from the Vistula Lagoon directly into the Baltic Sea at the beginning of this century; (b) the intensification of water exchange with the sea because of
As far as physics is concerned, the required background knowledge is by and large classical physi... more As far as physics is concerned, the required background knowledge is by and large classical physics. Its basis is long-lasting human experience with the surrounding physical world; it pertains to the properties of the space in which the physical objects exist; it is concerned with the processes which these physical objects undergo and searches for the reasons behind their occurrence. This experience must not depend on the observer; it should be objective and ought to be expressed in an observer-independent fashion, verified only by real physical experiments.
The first basic thoughts and experiments on turbulence are due to Reynolds [20] who studied the f... more The first basic thoughts and experiments on turbulence are due to Reynolds [20] who studied the flow of a fluid through pipes with circular cross-sections. He recognised (by adding dye through a pipette to the fluid) that, basically, two flow regimes exist. In one case, the so-called laminar flow , the dye forms a coherent thin filament; in the second case, known as turbulent flow , the dye filament is torn very quickly after it left the nozzle of the pipette and is spread over the entire cross-section of the pipe; Fig. 6.1.
In Chap. 7 of Volume I, the propagation of surface waves in a layer of a homogeneous fluid referr... more In Chap. 7 of Volume I, the propagation of surface waves in a layer of a homogeneous fluid referred to an inertial frame was studied. It was shown that superposing the fields of two waves, with the same frequency propagating in opposite directions with the same amplitude can be combined to a standing wave. These standing waves appear as localized oscillations between fixed nodal lines of which the distance defines the semi-wave length with wave humps and wave troughs arising inbetween. Under frictionless conditions imaginary walls can be placed at any position parallel to the wave direction to confine a channel without physically violating any boundary conditions. Similarly, the locations of the nodal lines across the channel turned out to be the positions of standing waves where the longitudinal velocity component vanishes for all time so that vertical walls can equally be inserted at these positions without disturbing the solution. This then formally yields the surface wave solution for the unidirectional motion in a basin of rectangular form and constant depth, see Figs. 7.9 and 7.12 in Chap. 7 of Volume I. These standing wave solutions were subsequently generalized to two-dimensional oscillations in rectangular cells of constant depth in which non-vanishing horizontal velocity components are allowed within the cell that only persistently vanish at the four side walls, thus forming oscillations of true cellular structure (see Figs. 7.14 and 7.15 in Chap. 7 in Volume I). How does the structure of these waves change when the fluid is rotating?
Chapter 4 was devoted to the derivation and presentation of the governing equations of fluid mech... more Chapter 4 was devoted to the derivation and presentation of the governing equations of fluid mechanics and thermodynamics as they apply to fluid bodies under motion. The intention was to build a basic understanding of the mathematical description of the physical laws of balances of mass, momenta and energy in a form sufficiently general to all situations which one could possibly encounter in applications of physical limnology needed for this book.
In parts A and B of the last chapter, two sources of higher order baroclinicity were looked at (1... more In parts A and B of the last chapter, two sources of higher order baroclinicity were looked at (1) a two-layer fluid system with a diffusive interface and (2) a three-layer configuration with two sharp interfaces due to the presence of a thermocline and a chemocline. In this chapter we give further field evidences of higher order baroclinicity. Both cases are to a certain extent idealized; in a realistic situation, density changes are generally less abrupt and should be represented by using a thermal equation of state ρ = ρ(T, s) from measured temperature and electrical conductivity profiles. If this argument is consistently adopted, this would, strictly, mean that a numerical model for a stratified lake should be based on a multi-layer model, e.g. with linear density variation across each layer. For reasons of accurate determination of the phase speeds of the higher baroclinic seiche, this should be done so, even if only fundamental (V1) and first higher order (V2) modes are of interest.
Traditionally water is managed by different geographical compartments (e.g. rivers, reservoirs, l... more Traditionally water is managed by different geographical compartments (e.g. rivers, reservoirs, lakes, estuaries, ground) using specific tools for each compartment and often by different institutions. When specific tools are used for each compartment, the interactions between compartments are specified through boundary conditions (e.g. aquifer recharge in case of aquifer management), which must be set by means of field data describing spatial and temporal variability. The specification of the boundary conditions between compartments into a watershed causes errors in hindcasting and makes forecasts difficult because data related to boundaries must be forecasted or generated. IWRM solves this difficulty because management tools for all the geographical compartments are coupled and used interactively. The implementation of an integrated approach is more complex, but its exploitation is much more economical, minimizing the amount of data required for running the models. Another advantage of the integrated approach is that the modelling system provides results to all the stakeholders in the catchment and consequently it stimulates the cooperation among them. In the framework of this cooperation it will be easier to share knowledge, data and working methods, with advantages in terms of investment and terms of solutions acceptance by stakeholders.
In Chaps. 7 and 8 of Volume I, an introduction was given to the mathematical treatment of linear ... more In Chaps. 7 and 8 of Volume I, an introduction was given to the mathematical treatment of linear waves in general and to water waves in particular. To isolate the specific properties of water waves with a free surface, the influence of the rotation of the reference frame was not considered. Here, our aim is to elucidate the role played by the rotation of the reference frame – the Earth – in the dynamics of large water masses such as ponds, lakes, and the ocean.
Lake physics cannot be described let alone understood without tailoring the statements in mathema... more Lake physics cannot be described let alone understood without tailoring the statements in mathematical expressions and deducing results from these. We now wish to lay down the mathematical prerequisites that are indispensable to reach quantitative results. A systematic presentation will not be given because it is assumed that the reader is (or once has been) familiar with the subjects and only needs to be reminded of knowledge that may be somewhat dormant. Let us begin with mathematics.
We have expressed the fundamental physical ideas – that mass, momentum and energy must be conserv... more We have expressed the fundamental physical ideas – that mass, momentum and energy must be conserved – in the form of mathematical equations (balance laws) and demonstrated that the balance of moment of momentum in its local expression for a continuum requests that the (Cauchy) stress tensor is symmetric, but beyond this does not produce any further local equation. So, it appears that the conservation law of moment of momentum is superfluous. This is not so; correct is that by requesting the Cauchy stress to be symmetric and exploiting pointwise the balances of mass, momentum and energy then automatically also guarantee the balance law of angular momentum to be identically satisfied. However, since physically, linear momentum is associated with the translational motion and angular momentum with the rotatory motion, the rotational behavior can often better be identified if the law of balance of angular momentum is explicitly employed.
Waves in open waters, such as the ocean, lakes, channels, arise in a variety of forms and types a... more Waves in open waters, such as the ocean, lakes, channels, arise in a variety of forms and types and have various physical reasons for their formation. We will have the occasion in a number of subsequent chapters (in volume 2) to investigate the important types of waves in the geophysical context as they apply to lakes, the ocean and, to limited extent, also to the atmosphere.
In this chapter the intention is to describe the vertical and (eventually) also horizontal struct... more In this chapter the intention is to describe the vertical and (eventually) also horizontal structure of the horizontal current in lakes which are subjected to external wind forces. The water will be assumed to be homogeneous or stratified in two layers, and the internal friction and the effects of the rotation of the Earth will play an important role in the establishment of the current distribution.
We study how a coastal obstruction (peninsula or coastal island) affects the three-dimensional ba... more We study how a coastal obstruction (peninsula or coastal island) affects the three-dimensional barotropic currents in an oblong rectangular basin with variable bathymetry across the basin width. The transverse depth profile is asymmetric and the peninsula or island lies in the middle of the long side of the rectangle. A semi-spectral model for the Boussinesq-approximated shallow water equations, developed in Haidvogel et al. and altered for semi-implicit numerical integration in time in Wang and Hutter, is used to find the steady barotropic state circulation pattern to external winds. The structural (qualitative) rearrangements and quanti2tative features of the current pattern are studied under four principal wind directions and different lengths of the peninsula and its inclination relative to the shore. The essentially non-linear relationships of the water flux between the two sub-basins (formed by the obstructing peninsula) and the corresponding cross-sectional area left open are found and analysed. It is further analysed whether the depth-integrated model, usually adopted by others, is meaningful when applied to the water exchange problems. The flow through the channel narrowing is quantitatively estimated and compared with the three-dimensional results. The dynamics of the vortex structure and the identification of the up-welling/down-welling zones around the obstruction are discussed in detail. The influence of the transformation of the peninsula into a coastal island on the global basin circulation is considered as are the currents in the channel. The geometric and physical reasons for the anisotropy of the current structure which prevail through all obtained solutions are also discussed.
The results of qualitative laboratory and numerical experiments on two-dimensional non-linear mod... more The results of qualitative laboratory and numerical experiments on two-dimensional non-linear model are described, aiming at an investigation of the structure of the front of bottom gravity current. Non-coincidence of frontal interfaces in density and velocity fields within the bottom gravity current leading (frontal) part is stated on the base of comparative analysis of numerical and laboratory experiments. This fact is experimentally confirmed by field example of marine water inflow into a brackish lagoon. The density gradient along the stream line is shown to be an additional effective criterion for the localization of the frontal zone.
The results of qualitative laboratory and numerical experiments on two-dimensional non-linear mod... more The results of qualitative laboratory and numerical experiments on two-dimensional non-linear model are described, aiming at an investigation of the structure of the front of bottom gravity current. Non-coincidence of frontal interfaces in density and velocity fields within the bottom gravity current leading (frontal) part is stated on the base of comparative analysis of numerical and laboratory experiments. This fact is experimentally confirmed by field example of marine water inflow into a brackish lagoon. The density gradient along the stream line is shown to be an additional effective criterion for the localization of the frontal zone.
Development of a thermal bar in a laboratory flume with an inclined bottom (3.7°–12°) under the c... more Development of a thermal bar in a laboratory flume with an inclined bottom (3.7°–12°) under the conditions of cooling/heating of the water with a temperature close to that of the maximal density is studied. The structure of the temperature field and currents during different stages of the circulation is examined: (i) formation of an along-slope gravity current, (ii) generation of a subsurface jet, and (iii) transformation of one type of the circulation into another at passing the temperature of the maximum density. The “fall” and “spring” types of the thermal bar are shown to be dynamically equivalent: the transport of the near-shore waters to the deepwater part, which is driven by the buoyancy flux rather than by the heat flux across the surface, transforms stage (i) into stage (ii), while the opposite (on-shore) flow is generated in the intermediate layers. A comparison of the results with the field and laboratory data published allows us to suggest that the propagation of the thermal bar front in the “fast” stage can be considered as the development of a convective jet with its velocity U ∼ h 3/4, which is proportional to the growing thickness of the upper layer h affected by the heating/cooling processes
We report the results of laboratory experiments on water heating/cooling, performed in 5 m long w... more We report the results of laboratory experiments on water heating/cooling, performed in 5 m long water channel with a slope. About 63 series of photos were analyzed: for 3 locations, for 3 bottom slopes (3.7, 6.7 and 12 degrees) and for different Ra numbers. It was pointed out that there exist two types of mixing characterizing different circulations in the presence of slope: gravity current and undersurface jet; the thermal bar is the region where one type of mixing is replaced by another; the highest speed and flowrate are at the break point; the flow is three-dimensional.
Development of a thermal bar in a laboratory flume with an inclined bottom (3.7° 12°) under the c... more Development of a thermal bar in a laboratory flume with an inclined bottom (3.7° 12°) under the conditions of cooling/heating of the water with a temperature close to that of the maximal density is studied. The structure of the temperature field and currents during different stages of the circulation is examined: (i) formation of an along-slope gravity current, (ii) generation of a subsurface jet, and (iii) transformation of one type of the circulation into another at passing the temperature of the maximum density. The “fall” and “spring” types of the thermal bar are shown to be dynamically equivalent: the transport of the near-shore waters to the deepwater part, which is driven by the buoyancy flux rather than by the heat flux across the surface, transforms stage (i) into stage (ii), while the opposite (on-shore) flow is generated in the intermediate layers. A comparison of the results with the field and laboratory data published allows us to suggest that the propagation of the thermal bar front in the “fast” stage can be considered as the development of a convective jet with its velocity U ˜ h 3/4, which is proportional to the growing thickness of the upper layer h affected by the heating/cooling processes
The man-made contribution to significant increase in salinity in the Vistula Lagoon (south-easter... more The man-made contribution to significant increase in salinity in the Vistula Lagoon (south-eastern Baltic) during the last century is discussed in this paper: (a) diversion of the main part of the Vistula River discharge from the Vistula Lagoon directly into the Baltic Sea at the beginning of this century; (b) the intensification of water exchange with the sea because of
As far as physics is concerned, the required background knowledge is by and large classical physi... more As far as physics is concerned, the required background knowledge is by and large classical physics. Its basis is long-lasting human experience with the surrounding physical world; it pertains to the properties of the space in which the physical objects exist; it is concerned with the processes which these physical objects undergo and searches for the reasons behind their occurrence. This experience must not depend on the observer; it should be objective and ought to be expressed in an observer-independent fashion, verified only by real physical experiments.
The first basic thoughts and experiments on turbulence are due to Reynolds [20] who studied the f... more The first basic thoughts and experiments on turbulence are due to Reynolds [20] who studied the flow of a fluid through pipes with circular cross-sections. He recognised (by adding dye through a pipette to the fluid) that, basically, two flow regimes exist. In one case, the so-called laminar flow , the dye forms a coherent thin filament; in the second case, known as turbulent flow , the dye filament is torn very quickly after it left the nozzle of the pipette and is spread over the entire cross-section of the pipe; Fig. 6.1.
In Chap. 7 of Volume I, the propagation of surface waves in a layer of a homogeneous fluid referr... more In Chap. 7 of Volume I, the propagation of surface waves in a layer of a homogeneous fluid referred to an inertial frame was studied. It was shown that superposing the fields of two waves, with the same frequency propagating in opposite directions with the same amplitude can be combined to a standing wave. These standing waves appear as localized oscillations between fixed nodal lines of which the distance defines the semi-wave length with wave humps and wave troughs arising inbetween. Under frictionless conditions imaginary walls can be placed at any position parallel to the wave direction to confine a channel without physically violating any boundary conditions. Similarly, the locations of the nodal lines across the channel turned out to be the positions of standing waves where the longitudinal velocity component vanishes for all time so that vertical walls can equally be inserted at these positions without disturbing the solution. This then formally yields the surface wave solution for the unidirectional motion in a basin of rectangular form and constant depth, see Figs. 7.9 and 7.12 in Chap. 7 of Volume I. These standing wave solutions were subsequently generalized to two-dimensional oscillations in rectangular cells of constant depth in which non-vanishing horizontal velocity components are allowed within the cell that only persistently vanish at the four side walls, thus forming oscillations of true cellular structure (see Figs. 7.14 and 7.15 in Chap. 7 in Volume I). How does the structure of these waves change when the fluid is rotating?
Chapter 4 was devoted to the derivation and presentation of the governing equations of fluid mech... more Chapter 4 was devoted to the derivation and presentation of the governing equations of fluid mechanics and thermodynamics as they apply to fluid bodies under motion. The intention was to build a basic understanding of the mathematical description of the physical laws of balances of mass, momenta and energy in a form sufficiently general to all situations which one could possibly encounter in applications of physical limnology needed for this book.
In parts A and B of the last chapter, two sources of higher order baroclinicity were looked at (1... more In parts A and B of the last chapter, two sources of higher order baroclinicity were looked at (1) a two-layer fluid system with a diffusive interface and (2) a three-layer configuration with two sharp interfaces due to the presence of a thermocline and a chemocline. In this chapter we give further field evidences of higher order baroclinicity. Both cases are to a certain extent idealized; in a realistic situation, density changes are generally less abrupt and should be represented by using a thermal equation of state ρ = ρ(T, s) from measured temperature and electrical conductivity profiles. If this argument is consistently adopted, this would, strictly, mean that a numerical model for a stratified lake should be based on a multi-layer model, e.g. with linear density variation across each layer. For reasons of accurate determination of the phase speeds of the higher baroclinic seiche, this should be done so, even if only fundamental (V1) and first higher order (V2) modes are of interest.
Traditionally water is managed by different geographical compartments (e.g. rivers, reservoirs, l... more Traditionally water is managed by different geographical compartments (e.g. rivers, reservoirs, lakes, estuaries, ground) using specific tools for each compartment and often by different institutions. When specific tools are used for each compartment, the interactions between compartments are specified through boundary conditions (e.g. aquifer recharge in case of aquifer management), which must be set by means of field data describing spatial and temporal variability. The specification of the boundary conditions between compartments into a watershed causes errors in hindcasting and makes forecasts difficult because data related to boundaries must be forecasted or generated. IWRM solves this difficulty because management tools for all the geographical compartments are coupled and used interactively. The implementation of an integrated approach is more complex, but its exploitation is much more economical, minimizing the amount of data required for running the models. Another advantage of the integrated approach is that the modelling system provides results to all the stakeholders in the catchment and consequently it stimulates the cooperation among them. In the framework of this cooperation it will be easier to share knowledge, data and working methods, with advantages in terms of investment and terms of solutions acceptance by stakeholders.
In Chaps. 7 and 8 of Volume I, an introduction was given to the mathematical treatment of linear ... more In Chaps. 7 and 8 of Volume I, an introduction was given to the mathematical treatment of linear waves in general and to water waves in particular. To isolate the specific properties of water waves with a free surface, the influence of the rotation of the reference frame was not considered. Here, our aim is to elucidate the role played by the rotation of the reference frame – the Earth – in the dynamics of large water masses such as ponds, lakes, and the ocean.
Lake physics cannot be described let alone understood without tailoring the statements in mathema... more Lake physics cannot be described let alone understood without tailoring the statements in mathematical expressions and deducing results from these. We now wish to lay down the mathematical prerequisites that are indispensable to reach quantitative results. A systematic presentation will not be given because it is assumed that the reader is (or once has been) familiar with the subjects and only needs to be reminded of knowledge that may be somewhat dormant. Let us begin with mathematics.
We have expressed the fundamental physical ideas – that mass, momentum and energy must be conserv... more We have expressed the fundamental physical ideas – that mass, momentum and energy must be conserved – in the form of mathematical equations (balance laws) and demonstrated that the balance of moment of momentum in its local expression for a continuum requests that the (Cauchy) stress tensor is symmetric, but beyond this does not produce any further local equation. So, it appears that the conservation law of moment of momentum is superfluous. This is not so; correct is that by requesting the Cauchy stress to be symmetric and exploiting pointwise the balances of mass, momentum and energy then automatically also guarantee the balance law of angular momentum to be identically satisfied. However, since physically, linear momentum is associated with the translational motion and angular momentum with the rotatory motion, the rotational behavior can often better be identified if the law of balance of angular momentum is explicitly employed.
Waves in open waters, such as the ocean, lakes, channels, arise in a variety of forms and types a... more Waves in open waters, such as the ocean, lakes, channels, arise in a variety of forms and types and have various physical reasons for their formation. We will have the occasion in a number of subsequent chapters (in volume 2) to investigate the important types of waves in the geophysical context as they apply to lakes, the ocean and, to limited extent, also to the atmosphere.
In this chapter the intention is to describe the vertical and (eventually) also horizontal struct... more In this chapter the intention is to describe the vertical and (eventually) also horizontal structure of the horizontal current in lakes which are subjected to external wind forces. The water will be assumed to be homogeneous or stratified in two layers, and the internal friction and the effects of the rotation of the Earth will play an important role in the establishment of the current distribution.
We study how a coastal obstruction (peninsula or coastal island) affects the three-dimensional ba... more We study how a coastal obstruction (peninsula or coastal island) affects the three-dimensional barotropic currents in an oblong rectangular basin with variable bathymetry across the basin width. The transverse depth profile is asymmetric and the peninsula or island lies in the middle of the long side of the rectangle. A semi-spectral model for the Boussinesq-approximated shallow water equations, developed in Haidvogel et al. and altered for semi-implicit numerical integration in time in Wang and Hutter, is used to find the steady barotropic state circulation pattern to external winds. The structural (qualitative) rearrangements and quanti2tative features of the current pattern are studied under four principal wind directions and different lengths of the peninsula and its inclination relative to the shore. The essentially non-linear relationships of the water flux between the two sub-basins (formed by the obstructing peninsula) and the corresponding cross-sectional area left open are found and analysed. It is further analysed whether the depth-integrated model, usually adopted by others, is meaningful when applied to the water exchange problems. The flow through the channel narrowing is quantitatively estimated and compared with the three-dimensional results. The dynamics of the vortex structure and the identification of the up-welling/down-welling zones around the obstruction are discussed in detail. The influence of the transformation of the peninsula into a coastal island on the global basin circulation is considered as are the currents in the channel. The geometric and physical reasons for the anisotropy of the current structure which prevail through all obtained solutions are also discussed.
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