Midterm: UNIT ONE Flashcards
(35 cards)
Which is Material view?
Which is Field view?
Lagrangian –> Material
Eulerian –> Field
What is the interest of material view
obeys Newton’s law and thermodynamics –> construct laws for the atmosphere and ocean
What is Rossby number and what does it imply?
scaling of acceleration versus coriolis force, if small –> can neglect acceleration
eg: geostrophic balance
What is the material derivative and what is each term contribution?
Eulerian: rate of change of a particular fluid element versus rate of change at a fixed point in space
advection: ability of the fluid to carry its properties as it moves/ contribution from spatial variation of the property experienced as the parcel moves.
What is the circulation equation?
int_c ( u dot dl ) = dint_surf ( curl of u )dot n dS –> Stokes theorem
Line integral along a closed curve C of the tangential velocity equals the surface integral over any surface bounded by C of the curl of the velocity normal to the surface
-Circulation to vorticity
Physical meaning of divergence
For the velocity field:
Measures how fast the flow is expanding at a given point
Parcel Volume may expand/contract as the parcel moves around in the atm/ocean. How does the volume change?
Changes through net inward/outward motion through the surface that encloses the cube
What is the fractional rate of change with time of the parcel’s volume?
1/delV d(delV)/dt
Derive Divergence
∇ ∙ u
-Expansion/contraction of the cube and the difference in advection speed on opposing faces of the cube
Divergence theorem
d_int_S [ u dot N ] dS = t_int_V [ ∇ ∙ u ] dV
the integral over the closed surface that contains V of the velocity normal to this surface = integral within volume V enclosed by the surface of the divergence of the velocity
What is Helmotz decomposition?
and express the horizontal flow through the gradients of the two potentials
Two components of the flow:
-divergence free: Stream function
-curl-free: Velocity Potential
–> flow can be resolved as a sum of the two potentials
𝑢 = − 𝜕Ψ/𝜕𝑦 + 𝜕χ/𝜕𝑥
𝑣 = 𝜕Ψ/𝜕𝑥 + 𝜕χ/𝜕𝑦
With what is the vertical vorticity associated with?
Entirely associated with potential Ψ, streamfunction
∇ℎ^2 Ψ
With what is the divergence associated with?
Entirely associated with χ, velocity potential
∇ℎ ∙ 𝒖 = ∇ℎ^2 χ
How do you get a non-divergent flow?
∇ℎ ∙ 𝒖 = ∇ℎ^2 χ = 0
𝑢 = − 𝜕Ψ/𝜕𝑦
𝑣 = 𝜕Ψ/𝜕𝑥
-streamfunction parallel with u, magnitude of u = ∇ℎ Ψ
How do you get irrotational flow?
vort_z = k(curl u) = ∇ℎ^2 Ψ = 0
𝑢 = 𝜕χ/𝜕𝑥
𝑣 = 𝜕χ/𝜕𝑦
-divergence velocity potential perpendicular to u , ||u|| = ||∇ℎχ||
Derive pressure gradient force
p = F/dS
F = (p1 - p2)dS = (p1=p2)delydelz
p1 = p(x_0 - delx/2,y_0,z_0) –> Taylor expansion in x
–> F1p - F2p = − 𝜕𝑝/𝜕𝑥 𝛿𝑥 𝛿𝑦 𝛿𝑧
In 3D –> F_p = −∇𝑝 𝛿𝑥𝛿𝑦𝛿𝑧
divide by mass – > F_p/unit mass = − 1/𝜌 ∇𝑝
What are the 3 forces acting on a parcel (in inertial frame of reference)?
Gravity : -g k
PGF: − 1/𝜌 ∇𝑝
Friction: F
Ftot/m = Du/Dt –> get momemtum equation
Derive principle of mass conservation
–> Continuity equation
1/𝛿𝑀 𝐷(𝛿𝑀)/𝐷𝑡 = 0
–> D𝑝/Dt + 𝑝∇ ∙ 𝒖 = 0 : Divergence form
Implication of incompressible flow in terms of continuity equation?
∇ ∙ 𝒖 = 0 –> non-divergent flow
So, can get information on vertical acceleration
∇h u = - 𝜕𝑤/𝜕𝑧 (unless all gradients are 0)
From divergence form of continuity equation, what is the mass flux form?
Express material derivative
–>
𝜕𝜌/𝜕𝑡 + 𝜵 ∙ 𝜌 ∙ 𝒖 = 0
What is the mass flux?
The rate at which mass enters the cube through the surface
Net mass flux through a surface is
(𝜌1𝑢1 − 𝜌2𝑢2)𝛿𝑦𝛿𝑧 –> Taylor expand to get that the net mass flux per unit volume is
-∇ ∙ 𝜌𝒖
Write the momentum equation in rotating frame
(𝑑𝒖_𝑅/𝑑𝑡)_𝑅 = (𝑑𝒖_𝐼/𝑑𝑡)_𝐼− 2Ω × 𝒖𝑅 − Ω × (Ω × 𝒓)
How can centrifugal force be rewritten?
Ω × (Ω × 𝒓) = Ω^𝟐𝒓¬
Given the dependence on r, the centrifugal acceleration may also be expressed as gradient of a potential:
∇Φ𝑐𝑒= (∇ 1/2 Ω^2𝑟¬^2)
Φ𝑐𝑒 = 1/2 Ω^2𝑟¬^2
Rewrite the momentum equation in rotating frame by replacing what the inertial acceleration is
(𝑑𝒖_𝑅/𝑑𝑡)_𝑅 = 1/𝜌 ∇𝑝 -gk − 2Ω × 𝒖_𝑅 + ∇Φ𝑐𝑒
