Gravity Currents and Intrusions: Analysis and Prediction

Contents
Preface
About the Author
Copyright acknowledgment
Abbreviations
1. Introduction
	1.1 The gravity current (GC)
	1.2 Classification
	1.3 Approximate models
		1.3.1 Some words of caution
2. Theoretical formulation
	2.1 The Navier–Stokes (NS) equations
	2.2 General thin-layer balances
		2.2.1 Motion of the interface and the continuity equation
		2.2.2 Hydrostatic pressure and buoyancy-driving term
			2.2.2.1 Buoyancy-driving term for constant ρa and ρc
			2.2.2.2 Non-constant ρa and/or ρc
PART I : NON-STRATIFIED AMBIENT
	3. Shallow-water (SW) analysis of the inertial GC: The one-layer model
		3.1 Governing equations
			3.1.1 Volume and momentum
			3.1.2 Nose jump conditions
			3.1.3 Some remarks
			3.1.4 Scaling
			3.1.5 Summary 1
			3.1.6 A useful transformation
			3.1.7 Summary 2
			3.1.8 Input parameters and the Boussinesq simplification
		3.2 The full behavior by numerical solution
		3.3 Dam-break and slumping stages
			3.3.1 The gate and reservoir (lock)
			3.3.2 Constant h = hN domain
		3.4 Similarity solution
			3.4.1 Fixed volume V
			3.4.2 Extensions for V = qtα
			3.4.3 Local similarity
			3.4.4 Non-Boussinesq systems
		3.5 Effective Ree of the SW solution and transition xV
			3.5.1 Non-Boussinesq extension
		3.6 Reservoir drainage from an edge
	4. Jumps in steady-state streams and control-volume matching conditions
		4.1 General considerations
		4.2 Internal jump solution
			4.2.1 Major result
			4.2.2 Vorticity balance
			4.2.3 Small amplitude jump h/hu → 1
			4.2.4 General cross-section
		4.3 Front jump and Benjamin’s analysis
			4.3.1 Major result Fr(a)
			4.3.2 Energy considerations: The amax restriction for Fr
			4.3.3 Summary of Benjamin’s solution
			4.3.4 Jump condition
			4.3.5 Generalization of Benjamin’s analysis
				4.3.5.1 Fixed top
				4.3.5.2 Open top
			4.3.6 “Critical” nose: The acrit restriction for Fr
				4.3.6.1 Fixed top
				4.3.6.2 Open top
			4.3.7 How to assess a given Fr formula
		4.4 Internal jump in one-layer model
		4.5 Jump reflected by a vertical boundary
	5. Box models for 2D geometry
		5.1 Fixed-volume current with inertial–buoyancy balance
		5.2 Inflow volume change V = qtα
			5.2.1 Inertial–buoyancy balance
			5.2.2 Viscous–buoyancy balance
			5.2.3 Transition length xV
			5.2.4 Critical α
			5.2.5 Comparisons and experimental verification
			5.2.6 Non-Newtonian power-law viscosity
		5.3 Extension to non-rectangular cross-section
	6. Two-layer SW model
		6.1 Introduction
		6.2 The governing equations
		6.3 Boussinesq system in dimensionless form
			6.3.1 “Critical” nose acrit restriction for H < 1.25
		6.4 Jumps of interface for H < Hcrit = 2
			6.4.1 Backward-moving discontinuity
			6.4.2 The reflected bore
		6.5 Dam break and speed of propagation during slumping
			6.5.1 Theory
			6.5.2 Some comparisons
			6.5.3 The slumping distance: Theory and experiments
		6.6 Energy and work in a two-layer model
			6.6.1 Some SW energy calculations
			6.6.2 Analytical global energy balances
				6.6.2.1 General observation: The dissipation effect of the jump
				6.6.2.2 The energy-conserving dilemma
			6.6.3 Energy in the similarity stage
			6.6.4 Conclusions
		6.7 More complex configurations
	7. Axisymmetric GC in inertial regime: SW analysis
		7.1 The governing equations
			7.1.1 Summary 1
		7.2 A useful transformation
			7.2.1 Summary 2
		7.3 The full behavior by numerical solution
		7.4 Dam-break stage
		7.5 Similarity solution
			7.5.1 Fixed volume V
			7.5.2 Extensions for V = qtα
			7.5.3 Local similarity
		7.6 Effective Ree of the SW solution and transition rV
		7.7 Some comparisons
		7.8 Reservoir drainage from an edge
		7.9 GC sustained by a constant source V = qt
	8. Box models for axisymmetric geometry
		8.1 Fixed-volume current with inertial–buoyancy balance
		8.2 Inflow volume change V = qtα
			8.2.1 Inertial–buoyancy balance
			8.2.2 Viscous–buoyancy balance
			8.2.3 Critical α
			8.2.4 Non-Newtonian power-law viscosity
		8.3 Hybrid model for V = qt (α = 1)
	9. Effects of rotation
		9.1 Introduction and the f/N ratio
		9.2 Axisymmetric (cylindrical) current
			9.2.1 The scaled SW equations and boundary conditions
				9.2.1.1 Potential vorticity
			9.2.2 Steady-state lens (SL)
				9.2.2.1 Behavior for C ≪ 1
				9.2.2.2 Behavior for C ≫ 1
				9.2.2.3 Aspect ratio of SL
			9.2.3 The SW current and formation of the lens
			9.2.4 Two-layer models and more about the lens
				9.2.4.1 Lens with source at origin
			9.2.5 Some experimental and NS results
				9.2.5.1 The current
				9.2.5.2 The lens
			9.2.6 Summary
		9.3 Laterally unbounded Cartesian 2.5D current
			9.3.1 The SW model
				9.3.1.1 Time-dependent major propagation
				9.3.1.2 Steady-state wedge
			9.3.2 Spin-up
			9.3.3 NS support
			9.3.4 Extensions: Finite width, double-wedge (xz-vein) and slope effect
		9.4 Rotating channel
			9.4.1 Steady-state results
			9.4.2 Dam-break considerations
	10. Buoyancy decays: Particle-driven GC, porous boundary, and entrainment
		10.1 Particle-driven currents
			10.1.1 Motion and concentration of particles
			10.1.2 The motion of the interface
			10.1.3 Effective reduced gravity and reversing buoyancy effect
			10.1.4 Momentum equation
			10.1.5 The governing SW equations
				10.1.5.1 Model T
				10.1.5.2 Model L
			10.1.6 Some SW results
				10.1.6.1 Sediment
			10.1.7 Similarity solution rudiments
			10.1.8 Box models
		10.2 Axisymmetric particle-driven current
			10.2.1 The governing SW equations
				10.2.1.1 Model T
				10.2.1.2 Model L
			10.2.2 Some SW results
				10.2.2.1 Sediment
			10.2.3 Similarity solution rudiments
			10.2.4 Box models
		10.3 Extensions of particle-driven solutions
		10.4 Current over a porous bottom
			10.4.1 The porous-boundary velocity condition
			10.4.2 The SW approximation
				10.4.2.1 The SW equations
				10.4.2.2 The global volume balance
				10.4.2.3 Some results
			10.4.3 Box models
		10.5 Axisymmetric current over a porous bottom
			10.5.1 The SW formulation
				10.5.1.1 The global volume balance
				10.5.1.2 Some results
			10.5.2 Box models
		10.6 Entrainment
	11. Inclined boundary effects
		11.1 Entrainment
		11.2 SW extended model with entrainment and drag
			11.2.1 Horizontal γ = 0 (with entrainment)
			11.2.2 Downslope current γ > 0
				11.2.2.1 Fixed volume
				11.2.2.2 Fixed influx, downslope current
				11.2.2.3 Summary
		11.3 Upslope currents γ < 0
		11.4 Viscous regime
	12. Non-Boussinesq systems: The one-layer SW model
		12.1 Introduction
		12.2 Formulation
		12.3 Dam-break and initial slumping motion
			12.3.1 Asymptotes for ρa/ρc → 0 and ρc/ρa → 0
			12.3.2 General results
		12.4 The transition and self-similar stages
		12.5 Summary
	13. Non-Boussinesq systems: The two-layer fixed-top SW model
		13.1 Introduction
		13.2 Formulation
		13.3 The heavy-fluid current domain
			13.3.1 Left-moving jump
				13.3.1.1 Restriction on the left-moving jump
			13.3.2 The reflected bore
			13.3.3 Critical nose speed and height
		13.4 The light-fluid current
			13.4.1 Right-moving free jump
				13.4.1.1 Restriction on the right-moving jump
			13.4.2 Critical nose speed and height
		13.5 The full-depth lock H = 1
		13.6 H > 1 cases
		13.7 Concluding remarks
		13.8 Asymptotic behavior for R → 0
			13.8.1 Heavy-fluid domain
			13.8.2 Light-fluid domain (air cavity)
		13.9 Tailwaters current
			13.9.1 The nose condition
			13.9.2 Method of solution
			13.9.3 Some results
			13.9.4 One-layer reduction
	14. Lubrication theory formulation for viscous currents
		14.1 2D geometry
			14.1.1 The governing equations
				14.1.1.1 Balances
				14.1.1.2 Boundary conditions at the nose
				14.1.1.3 Scaling consideration
			14.1.2 Similarity solution
			14.1.3 Summary
			14.1.4 Some comparisons
			14.1.5 Extension to surface currents and intrusions
		14.2 Axisymmetric current
			14.2.1 The governing equations
			14.2.2 Similarity solution
			14.2.3 Summary
			14.2.4 Some comparisons
		14.3 Non-Newtonian power-law viscosity
			14.3.1 2D case
				14.3.1.1 Experiments
			14.3.2 Axisymmetric case
				14.3.2.1 Experiments
		14.4 Two-layer model for viscous GC
		14.5 Non-rectangular cross-section
		14.6 Current in a porous medium
	15. Vegetation effects
		15.1 Theoretical model
		15.2 Similarity solutions
		15.3 One-layer solutions
			15.3.1 α = 2 : Linear H(y) and constant uN
			15.3.2 Fixed volume (α = 0) with linear U(y)
			15.3.3 Constant h(x = 0) current (δ = 0)
			15.3.4 Solutions for α ≠ 2, 0 and δ ≠ 0
		15.4 The Boussinesq lock-exchange problem
			15.4.1 λ = 1, β = 1/2
			15.4.2 λ = 2, β = 2/3
		15.5 The axisymmetric case, one-layer model
		15.6 A remark about c and cD
PART II : STRATIFIED AMBIENT CURRENTS AND INTRUSIONS
	16. Continuous density transition
		16.1 Introduction
		16.2 The SW formulation
			16.2.1 The nose condition
		16.3 SW results and comparisons with experiments and simulations
		16.4 Dam break
		16.5 Critical speed and nose–wave interaction
		16.6 Similarity solution
		16.7 The validity of the inertial (inviscid) approximation
		16.8 Double stratification (in both current and ambient)
			16.8.1 SW formulation
				16.8.1.1 Summary
			16.8.2 Dam-break results
			16.8.3 Time-dependent SW results
			16.8.4 Concluding remarks
	17. Axisymmetric and rotating cases
		17.1 SW formulation
			17.1.1 Steady-state lens (SL)
				17.1.1.1 Analytical approximations for the SL
				17.1.1.2 Analytical approximations for the SL with S = 1
				17.1.1.3 Numerical solution of the SL
			17.1.2 The energy of the SL
		17.2 SW and NS finite-difference results
			17.2.1 Sustained rotating intrusion
		17.3 The validity of the inertial (inviscid) approximation
	18. The steady-state current
		18.1 Steady-state flow pattern
		18.2 Results
			18.2.1 Small S (small γ)
			18.2.2 Large γ. Validity–stability and criticality
			18.2.3 The effective g′ and Fr
			18.2.4 Energy dissipation
		18.3 Comparisons and conclusions
	19. Intrusions in 2D geometry
		19.1 Introduction
		19.2 Two-layer stratification
		19.3 Linear transition layer
			19.3.1 Formulations
			19.3.2 SW equations and boundary conditions
		19.4 Rectangular lock configurations
			19.4.1 Part-depth transition layer and full-depth lock
			19.4.2 Fully linearly-stratified tank, part-depth locks
				19.4.2.1 Comparisons of SW, NS, and experiments
				19.4.2.2 Nose–wave interaction
		19.5 Cylindrical lock in a fully linearly-stratified tank
		19.6 Similarity solution
			19.6.1 Local similarity
		19.7 The validity of the inertial (inviscid) approximation
		19.8 Non-symmetric intrusions
			19.8.1 Equilibrium intrusion
			19.8.2 Full-depth lock
			19.8.3 Continuous linear stratification
			19.8.4 Summary
	20. Intrusions in axisymmetric geometry
		20.1 Introduction
		20.2 Two-layer stratification
		20.3 Fully linearly-stratified tank, part-depth locks
			20.3.1 Formulation and SW approximations
			20.3.2 Similarity solution
			20.3.3 Local similarity
			20.3.4 SW results and comparisons with NS
				20.3.4.1 The tail and variables stretching at long time
				20.3.4.2 Comparison to NS simulation
			20.3.5 The validity of the inertial approximation
			20.3.6 Summary
	21. Box models for 2D geometry
		21.1 Fixed volume and inertial–buoyancy balance
		21.2 S = 1, inflow volume change V = qtα
			21.2.1 Inertial–buoyancy balance
			21.2.2 Viscous–buoyancy balance
			21.2.3 Critical α
	22. Box models for axisymmetric geometry
		22.1 Fixed volume and inertial–buoyancy balance
		22.2 S = 1, inflow volume change V = qtα
			22.2.1 Inertial–buoyancy balance
			22.2.2 Viscous–buoyancy balance
			22.2.3 Critical α
			22.2.4 Some experimental support
	23. Hybrid model for sustained axisymmetric intrusion
		23.1 Non-rotating system
			23.1.1 The steady-state domain (ri < r < rJ(t))
			23.1.2 The head region (rJ(t) < r < rN(t))
				23.1.2.1 Long-time asymptotes
			23.1.3 Comparison of hybrid model to SW results
		23.2 Rotating frame
			23.2.1 SW components
				23.2.1.1 Steady states
				23.2.1.2 Temporal evolution: Numerical SW computations
			23.2.2 The Coriolis Hybrid model
				23.2.2.1 The head “box”
				23.2.2.2 First stage
				23.2.2.3 Second stage
				23.2.2.4 Third stage: The Coriolis lens
			23.2.3 Comparisons of hybrid model to SW results
			23.2.4 Conclusions
	24. Lubrication theory for viscous GCs with S = 1
		24.1 2D geometry
			24.1.1 Summary
		24.2 Axisymmetric geometry
			24.2.1 Summary
			24.2.2 Some comparisons
	25. Energy
		25.1 Introduction
		25.2 2D geometry
			25.2.1 SW formulation
				25.2.1.1 Energy and work
			25.2.2 NS considerations
			25.2.3 Results
		25.3 Axisymmetric geometry
			25.3.1 SW model
				25.3.1.1 Energy and work
			25.3.2 NS considerations
			25.3.3 Results
PART III : EXTENSIONS
	26. Non-rectangular cross-section area
		26.1 Introduction
		26.2 Two-layer SW model for fixed top
			26.2.1 Formulation
			26.2.2 Fr interlude
			26.2.3 Dimensionless variables and Re
			26.2.4 The components of the flow-field solution
				26.2.4.1 Characteristics and types of flow field
				26.2.4.2 Internal jump
			26.2.5 Critical nose speed and height
			26.2.6 Remark on shapes, types, and open top
			26.2.7 Example of dam-break solution
			26.2.8 The reflected bore
			26.2.9 Top current
		26.3 One-layer Bq models
			26.3.1 Formulation
			26.3.2 The slumping stage
			26.3.3 The general time-dependent flow
			26.3.4 Similarity solution
				26.3.4.1 The constant-influx case
			26.3.5 Particle-driven current
		26.4 Viscous regime
			26.4.1 Typical flow
			26.4.2 Transition xV
			26.4.3 Influx V = qtδ and critical δ
		26.5 Experimental support
		26.6 Concluding remarks
		26.7 Box model
			26.7.1 Constant ρc current
			26.7.2 Particle-driven current
	27. SW model extensions: Flux and open top
		27.1 Throughput flux in channel with fixed top
			27.1.1 Dimensionless variables
			27.1.2 The essentials of the flowfield solution
			27.1.3 Some results
		27.2 Open upper surface
			27.2.1 Lock-release two-layer flow
			27.2.2 One-layer reduction
		27.3 Conclusion
Appendix A Scaling suggestions
Appendix B SW equations: Characteristics and finite-difference schemes
	B.1 Characteristics
	B.2 Numerical solution of the SW equations
Appendix C NS numerical simulations
	C.1 Formulation
	C.2 A finite-difference code
	C.3 Other codes
Appendix D Some useful formulas
	D.1 Leibniz’s Theorem
	D.2 Vectors and coordinate systems
		D.2.1 Cartesian coordinates x, y, z
		D.2.2 Cylindrical coordinates r, θ, z
Notation guide
Bibliography
Index