PHYSICS OF LAKES: FOUNDATION OF THE MATHEMATICAL AND PHYSICAL BACKGROUND

By KAREN LEE

CONTENTS INTRODUCTION MOTIVATION LAKES ON EARTH LAKES CHARACTERISED BY THEIR RESPONSE TO THE DRIVING ENVIRONMENT SEASONAL CHARACTERISTICS CHARACTERISTICS BY MIXING BOUNDARY-RELATED PROCESSES CHARACTERISATION BY TYPICAL SCALES REFERENCES MATHEMATICAL PREREQUISITES SCALARS AND VECTORS TENSORS FIELDS AND THEIR DIFFERENTIATION GRADIENT, DIVERGENCE AND ROTATION OF VECTOR AND TENSOR FIELDS INTEGRAL THEOREMS OF VECTOR ANALYSIS GAUSS THEOREMS STOKES THEOREMS A BRIEF REVIEW OF THE BASIC THERMOMECHANICAL LAWS OF CLASSICAL PHYSICS UNDERLYING FUNDAMENTALS – GENERAL BALANCE LAWS PHYSICAL BALANCE LAWS BALANCE OF MASS BALANCE OF LINEAR MOMENTUM BALANCE OF MOMENT OF MOMENTUM BALANCE OF ENERGY SECOND LAW OF THERMODYNAMICS FUNDAMENTAL EQUATIONS OF LAKE HYDRODYNAMICS KINEMATICS BALANCES OF MOMENTUM AND MOMENT OF MOMENTUM, CONCEPT OF STRESS, HYDROSTATICS STRESS TENSOR LOCAL BALANCE LAW OF MOMENTUM OR NEWTON’S SECOND LAW MATERIAL BEHAVIOUR HYDROSTATICS BALANCE OF ENERGY: FIRST LAW OF THERMODYNAMICS DIFFUSION OF SUSPENDED SUBSTANCES SUMMARY OF EQUATIONS A FIRST LOOK AT THE BOUSSINESQ AND SHALLOW-WATER EQUATIONS CONSERVATION OF ANGULAR MOMENTUM–VORTICITY CIRCULATION SIMPLE VORTICITY THEOREMS HELMHOLTZ VORTICITY THEOREM POTENTIAL VORTICITY THEOREM TURBULENCE MODELLING A PRIMER ON TURBULENT MOTIONS AVERAGES AND FLUCTUATIONS FILTERS ISOTROPIC TURBULENCE REYNOLDS VERSUS FAVRE AVERAGES BALANCE EQUATIONS FOR THE AVERAGED FIELDS AVERAGING PROCEDURE AVERAGED DENSITY FIELD ??? DISSIPATION RATE DENSITY ??? REYNOLDS STRESS HYPOTHESIS ONE- AND TWO-EQUATION MODELS K–? MODEL FOR DENSITY-PRESERVING AND BOUSSINESQ FLUIDS THE BALANCE EQUATIONS CLOSURE RELATIONS SUMMARY OF (K ??)-EQUATIONS BOUNDARY CONDITIONS FINAL REMARKS HIGHER ORDER RANS MODELS LARGE EDDY SIMULATION AND DIRECT NUMERICAL SIMULATION EARLY ANISOTROPIC CLOSURE SCHEMES INTRODUCTION TO LINEAR WAVES THE LINEAR WAVE EQUATION AND ITS PROPERTIES SURFACE GRAVITY WAVES WITHOUT ROTATION SHORT-WAVE APPROXIMATION LONG-WAVE APPROXIMATION STANDING WAVES – RE?ECTION FREE LINEAR OSCILLATIONS IN RECTANGULAR BASINS OF CONSTANT DEPTH CONCLUDING REMARKS THE ROLE OF THE DISTRIBUTION OF MASS WITHIN WATER BODIES ON EARTH PROCESSES OF SURFACE WATER PENETRATION TO DEPTH HOMOGENISATION OF WATER MASSES REQUIRES ENERGY CONSTANT DENSITY LAYERS CONTINUOUS DENSITY VARIATION IN?UENCE OF THE THERMAL EXPANSION MOTION OF BUOYANT BODIES IN A STRATI?ED STILL LAKE IN?UENCE OF FRICTION INTERNAL OSCILLATIONS – THE DYNAMICAL IMPRINT OF THE DENSITY STRUCTURE FUNDAMENTAL EQUATIONS EIGENVALUE PROBLEM FOR THE VERTICAL MODE STRUCTURE IN CONSTANT DEPTH BASINS CLOSURE VERTICAL STRUCTURE OF WIND-INDUCED CURRENTS IN HOMOGENEOUS AND STRATI?ED WATERS PREVIEW AND SCOPE OF THIS CHAPTER HYDRODYNAMIC EQUATIONS APPLIED TO A NARROW LAKE UNDER STEADY WIND WIND-INDUCED STEADY CIRCULATION IN A NARROW HOMOGENEOUS LAKE OF CONSTANT DEPTH IN?UENCE OF BOTTOM SLIP ON THE WIND-INDUCED CIRCULATION WIND-INDUCED STEADY CIRCULATION IN A NARROW LAKE STRATI?ED IN TWO LAYERS EKMAN THEORY AND SOME OF ITS EXTENSIONS EKMAN SPIRAL STEADY WIND-INDUCED CIRCULATION IN A HOMOGENEOUS LAKE ON THE ROTATING EARTH WIND-DRIVEN STEADY CURRENTS IN LAKE ERIE TIME-DEPENDENT WIND-INDUCED CURRENTS IN SHALLOW LAKES ON THE ROTATING EARTH THE DYNAMICAL PREDICTION OF WIND TIDES ON LAKE ERIE PHENOMENOLOGICAL COEF?CIENTS OF WATER DENSITY OF WATER NATURAL WATER AND SEA WATER SUSPENDED MATTER SPECI?C HEAT OF WATER SPECI?C HEAT OF SALTY WATER VISCOSITY OF WATER PURE WATER SEA WATER NATURAL WATER MOLECULAR HEAT CONDUCTIVITY OF WATER HEAT CONDUCTIVITY OF SALT WATER IMPURITIES