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
