4. Shallow Layer Dynamics Flashcards
Shallow-Layer Dynamics
Outline
- simplest to consider coupled gravitational and rotational dynamics in a shallow-layer system (as opposed to stratification)
- assume a single step in density
- e.g. free water surface, in water ρ=const., in air ρ~0
Shallow-Layer Dynamics
Diagram
- neutral flat level of water surface is z=0
- rigid, uniform ocean bed at z=-H
- perturbation of the free surface z=h(x,y,t) which allows for |h|~|H|
- ‘wavelength’=L
- assume |h’| ~ H/L «_space;1
- horizontal velocity scale, u,v~U
Shallow-Water Dynamics
Governing Equations
-from incompressibility, W ~ HU/L << U so can disregard Dw/Dt => Du/Dt - fv = -1/ρ ∂p/∂x Dv/Dt + fu = -1/ρ ∂p/∂y 0 = -1/ρ ∂p/∂z - g ∂u/∂x + ∂v/∂y + ∂w/∂z = 0
Shallow Water Dynamics
Hydrostatic Balance
p = -ρg(z - h(x,y,t)) + po
- where the constant of integration is such that p=po at z=h
- this determines p everywhere
- can sub this back into the governing equations to eliminate p
Shallow-Water Dynamics
Boundary Conditions on h
-the kinematic condition states that fluid elements on the surface remain on the surface:
D/Dt (z-h(x,y,t)) = 0
-this is equivalent to no penetration at the surface
-the same as:
w = Dh/Dt at z=h
i.e.
w(x,h) = Dh/Dt
Shallow-Water Dynamics
Linearised Shallow-Water Equations
-linearise governing equations D/Dt derivatives become ∂/∂t
∂u/∂t - fv = -g ∂h/∂x
∂v/∂t + fu = -g ∂h/∂y
∂h/∂t = -H(∂u/∂x + ∂v/∂y)
Shallow-Water Dynamics
f=0
-differentiating the h equation with respect to t:
∂²h/∂t² - gH∇²h = 0
-c.f. sound waves equivalent wave equation
Shallow-Water Dynamics
f≠0
-introduces a new phenomenon
-this makes the linearised shallow-water equations a neat framework to study rotational-gravitational phenomenon
-if inertial terms are neglected:
-fv = -g ∂h/∂x
fu = -g ∂h/∂y
-the equations of geostrophic balance!!
Inertia-Gravity Waves
Derivation
-try a multivariable wave ansatz for u, v and h in the linearised shallow-water equations
Inertia-Gravity Waves
Dispersion Relation
ω[ω² - f² - gH(k² + l²)] = 0 -so have ω=0, geostrophic balance OR ω² = f² - gH(k² + l²)
Inertia-Gravity Waves
f=0
f=0 => ω = ± √[gH] k -gravity waves, governing equation: ∂²h/∂t² - gH∇²h = 0 -recovers gravity waves
Inertia-Gravity Waves
g=0
g=0 => ω = ±f -frequency of waves is the same as the Coriolis frequency -recovers inertial waves
Kelvin Waves
Outline
- same physics as inertial-gravity waves but different
- consider a coast with v=0 everywhere
Kelvin Waves
Governing Equations
-take linearised shallow-water equations and sub in v=0 => -inertial balance: ∂u/∂t = -g ∂h/∂x -geostrophic balance: fu = -g ∂h/∂y ∂h/∂t = -H ∂u/∂x -give two possibilities, evanescent (exponential) and sinusoidal
Kelvin Waves
Dispersion Relation
-try wave ansatz with y-structure function to be determined in the Kelvin wave governing equations
=>
ω = ±ck, where c = √[gH]
Kelvin Waves
h^
h^ = ∓ fc/g h^’
-since y≥0, need h^ not tending to infinity as y->∞
Kelvin Waves
Northern Hemisphere
f = fo > 0
=>
h^ = A exp(∓ fc/g y)
-if ω=-ck, h^ = A exp(+ fc/g y) -> ∞ unavoidably, need to exclude this solution
-if ω=+ck, h^= A exp(- fc/g y) which decays as y->∞
cp = ω/l = c
-waves propagate to the right
Kelvin Waves
Southern Hemisphere
f = fo < 0
-waves propagate to the left
Kelvin Waves
Equator
fo=0 -so need β-plane f = βy -get a different result for h^, a Gaussian not an exponential -waves propagate east
Potential Vorticity
Description
- consider a fluid column in a flow that spins at the same rate it is advected
- shallow-water system
Potential Vorticity
Governing Equation
-absolute vorticity:
q = f(y) + ζ
-in shallow-water systems, vortex stretching operates:
Dq/Dt = q_ . ∇ω_ ≠ 0
Potential Vorticity
Definition
-although q is not conserved, a related quantity is:
Q = q / [H+h]
-this represents the total angular momentum of a fluid column in a shallow water system
-angular momentum is conserved and Q is materially conserved
Rossby Adjustment
Description
- time-dependent adjustment of a shallow-water system from a geostrophic imbalance
- initial disturbance in h say, how does this evolve?
Rossby Adjustment
Governing Equations
-focus on linearised shallow-water equations with f=fo
-and the linearised form of DQ/Dt=0
=>
∂/∂t (ζ - hf/H) = 0
-where S(x,t)=ζ-hf/H represents linearised Q
-since ∂S/∂t=0, S is a constant, S=So(x) set by the initial conditions
Rossby Adjustment
Final Steady State
-suppose initial conditions h=ho(x), u=uo(x), v=vo(x) in 1D
-consider ∂/∂y=0
-focus on the final steady state, so ∂/∂t=0
=>
d²h/dx² - h/L² = f/g So(x)
-and
L² = gH/f²
Wind-Driven Ocean Circulation
Description
- imagine an ocean basin
- model using linearised shallow-water equations and assuming steady flow
Wind-Driven Ocean Circulation
Governing Equations
-linearised shallow water equations
-fv = -g ∂h/∂x - ku + F(y)
-where -fv is the Coriolis force, -g ∂h/∂x is gravity, -ku is drag from the ocean bed and F(y) is the wind drive
fu = -g ∂h/∂y - kv
-and mass conservation:
∂u/∂x + ∂v/∂y = 0
Wind-Driven Ocean Circulation
k=0
-sub k=0 into governing equations
-need β-plane approximation for f or get ∂F/∂y=0
f = fo + βy
=>
v = -1/β ∂F/∂y
-this implies the wind directly => v
-but for a closed basin, to conserve mass, we have;
∫v dx = 0
-where the integral is across the width of the basin from x=0 to x=a
-a contradiction
Wind-Driven Ocean Circulation
k>0
- βv = ∂F/∂y + k (∂v/∂x - ∂u/∂y)
- 2D incompressibility holds so can introduce streamfunction
- anticipate boundary layer to solve ∫v dx = 0 problem
Non-Linear Shallow-Water Dynamics
Examples
- tidal bore
- turbidity current -> sediment / particle laden flow in ocean caused e.g. by mudslide
- pyroclastic flow
- avalanches
Non-Linear Shallow-Water Dynamics
Governing Equations
-recall the non-linear shallow-water equations:
–mass conservation
∂h/∂t + ∂/∂x[hu] = 0
–momentum conservation
∂u/∂t + u ∂u/∂x = -g ∂h/∂x
-where h is now used to denote the full thickness, not the perturbed height as used for the linearised case
Non-Linear Shallow-Water Dynamics
Steady Flow
-for steady flow, ∂/∂x[hu]=0 so: hu = Q = const. -and from the governing equations: 1/2 u² + gh = K -sub in => 1/2 Q²/h² + gh = K
Non-Linear Shallow-Water Dynamics
Theory of Shocks
- non-linear shallow water systems naturally steepen
- in many situations (e.g. tidal bores) generating a shock, a region with an effectively vertical, continuously overturning a ‘hydraulic jump’
- turns out we can develop a direct theory for the evolution of xs(t), the shock position
Non-Linear Shallow-Water Dynamics
First Rankine-Hugonist Relation
-as ε->0
-xs’[h]± + [hu]± = 0
=>
xs’[h]± = [hu]±
-where ε is a small region either side of the shock front, xs
Non-Linear Shallow-Water Dynamics
Second Rankine-Hugonist Relation
-start with the governing equation in the x direction ∂u/∂t + u ∂u/∂x = -g ∂h/∂x -write in conservative form -integrate wrt x between xs-ε and xs+ε => xs' [hu]± = [hu² + 1/2 gh²]±
Gravity Currents
Description
-a shock coupled to time-dependent evolution of a shallow-water system for 0≤x≤xn(t)
Gravity Currents
Governing Equations
-start with the non-linear shallow water equations:
∂h/∂t + ∂/∂x[hu] = 0
∂u/∂t + u ∂u/∂x = -g ∂h/∂x
Gravity Currents
Steps
-apply the first Rankine-Hugoniot relation
-apply momentum balance at the front
-integrate from xn-ε to xn+ε
-replace with xn’=un
=>
1/2 u² = gh at x=xn(t)
-the so called Froude-number condition
-treat this as a boundary condition on the nonlinear shallow-water equations
Gravity Currents
Froude Number
-the Froude number:
Fr = √[u²/gh]
-at the front, Fr=√2
