SLIDE 1 A Test of Local Effective Diffusivity Parameterization in a Two-Layer, Wind-Driven Isopycnal Primitive Equation Model Yue-Kin Tsang
Center for Atmosphere Ocean Science Courant Institute of Mathematical Sciences New York University
SLIDE 2 Means and Eddies
Low-pass filtering via running average in space and time: ¯ f(x, y, t) = 1 τℓ2 t+τ/2
t−τ/2
dt′ x+ℓ/2
x−ℓ/2
dx′ y+ℓ/2
y−ℓ/2
dy′f(x′, y′, t′) Eddy components: f ′ ≡ f − ¯ f Averaging the passive tracer advection-diffusion equation ct + ∇ · (uc) = κ∇2c + S gives the large-scale equation ¯ ct + ∇ · (F + ¯ u¯ c) = κ∇2¯ c + ¯ S Eddy tracer flux: F = u′c′
SLIDE 3
Eddy Diffusivity
Flux-gradient relation: F = u′c′ = −K K K∇¯ c, where K K K = Kxx Kxy Kyx Kyy Eddy diffusion for ¯ c, ¯ ct + ∇ · (¯ u¯ c) = ∇ · [ (κI I I − K K K)∇¯ c ] + ¯ S If the small-scale statistics is inhomogeneous, K K K = K K K(x) Try to identify approximately local homogeneous regions (cells) and make estimation to K K K that is constant in a cell
SLIDE 4
Ocean Circulation Model
Two-layer, adiabatic, isopycnal primitive equations model (HIM) Mid-latitude, flat bottom basin of size 22◦ × 20◦ Zonal sinusoidal wind-forcing (double-gyre configuration) Linear bottom drag and biharmonic viscosity Resolution: 1/20 degree
SLIDE 5
Snapshots of u and v
u v
SLIDE 6 Example of Averaging
30 35 40 45 50
latitude
0.0 0.2 0.4 0.6 0.8
u u
along longitude=2.45
SLIDE 7
Snapshots of u′ and v′
u′ v′
SLIDE 8 PDF of u′ and v′
divide the domain into 2◦ × 2◦ cells
−0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.2 0.2 10
−5
10 −1 1 10
−5
10 −1 1 10
−5
10 −1 1 10
−5
10 −0.5 0.5 10
−5
10 −0.2 0.2 10
−5
10 −0.2 0.2 10
−5
10 −1 1 10
−5
10 −1 1 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.2 0.2 10
−5
10 −1 1 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
−5
10 −0.2 0.2 10
−5
10 −0.5 0.5 10
−5
10 −0.5 0.5 10
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10 −0.5 0.5 10
−5
10 −0.2 0.2 10
−5
10 −0.2 0.2 10
−5
10 −0.1 0.1 10
−5
10
SLIDE 9
Questions
Can we “measure” K
K K directly from high resolution
data? Can we make theoretical prediction on K
K K?
SLIDE 10
Measuring K K K(x)
Recall u′c′ = −K K K∇¯ c, write G = ∇¯ c: u′c′ = K K KxxGx + K K KxyGy v′c′ = K K KyxGx + K K KyyGy ⇒ four unknowns and two equations. K K K is property of flow, use two independent tracers a and b forced by different large-scale gradients: Γ = ∇¯ a, Λ = ∇¯ b, u′a′ = K K KxxΓx + K K KxyΓy v′a′ = K K KyxΓx + K K KyyΓy u′b′ = K K KxxΛx + K K KxyΛy v′b′ = K K KyxΛx + K K KyyΛy ⇒ four unknowns and four equations.
SLIDE 11 Tracer a and b
a b
5 10 15 20 −0.5 0.0 0.5 1.0 30 35 40 45 50 0.0 0.2 0.4 0.6 0.8 1.0
longitude (x) latitude (y)
SLIDE 12
Measured K K K(x)
K K Kxx K K Kxy K K Kyx K K Kyy
SLIDE 13 Transport Theory
Isotropic local mixing length estimate
VT =
Lmix = VT/|∇VT| K K Kxx = K K Kyy = cVTLmix
Shear dispersion in x + Mixing length in y
K K Kyy = cVTLjet (Ljet ≈ jet width) K K Kxx = U 2
jetL2 jet
K K Kyy (Ujet ≈ jet speed)
- K. S. Smith, J. Fluid Mech. 544, 133 (2005)
SLIDE 14
Isotropic Mixing Length Estimate
K K Kxx (or K K Kyy)
SLIDE 15
Summary
Study the eddy diffusivity tensor K
K K(x) in an
inhomogeneous system (the double-gyre configuration) Measure K
K K(x) from high resolution simulation data
Make theoretical estimation on K
K K(x) in some
regions of the domain using mixing length theory Work in progress: improve measurement of K
K K(x)
use more sophisticated theory to predict K
K K(x) in
the whole domain
