A Test of Local Effective Diffusivity Parameterization in a - PowerPoint PPT Presentation

A Test of Local Effective Diffusivity Parameterization in a Two-Layer, Wind-Driven Isopycnal Primitive Equation Model Yue-Kin Tsang K. Shafer Smith Center for Atmosphere Ocean Science Courant Institute of Mathematical Sciences New York University

Means and Eddies Low-pass filtering via running average in space and time: � t + τ/ 2 � x + ℓ/ 2 � y + ℓ/ 2 1 ¯ dt ′ dx ′ dy ′ f ( x ′ , y ′ , t ′ ) f ( x, y, t ) = τℓ 2 t − τ/ 2 x − ℓ/ 2 y − ℓ/ 2 Eddy components: f ′ ≡ f − ¯ f Averaging the passive tracer advection-diffusion equation c t + ∇ · ( u c ) = κ ∇ 2 c + S gives the large-scale equation c + ¯ c ) = κ ∇ 2 ¯ c t + ∇ · ( F + ¯ ¯ u ¯ S Eddy tracer flux: F = u ′ c ′

Eddy Diffusivity Flux-gradient relation:    K xx K xy F = u ′ c ′ = − K K K ∇ ¯ c, where K K K =  K yx K yy Eddy diffusion for ¯ c , c ] + ¯ c t + ∇ · (¯ ¯ u ¯ c ) = ∇ · [ ( κ I I − K K ) ∇ ¯ S I K 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

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

Snapshots of u and v u v

Example of Averaging o along longitude=2.45 u 0.8 u 0.6 0.4 0.2 0.0 -0.2 30 35 40 45 50 latitude

Snapshots of u ′ and v ′ u ′ v ′

PDF of u ′ and v ′ divide the domain into 2 ◦ × 2 ◦ cells 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.2 0 0.2 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −1 0 1 −1 0 1 −1 0 1 −0.5 0 0.5 −0.2 0 0.2 −0.2 0 0.2 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −1 0 1 −1 0 1 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.2 0 0.2 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −1 0 1 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.2 0 0.2 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.2 0 0.2 −0.2 0 0.2 −0.1 0 0.1

Questions Can we “measure” K K K directly from high resolution data? K Can we make theoretical prediction on K K ?

Measuring K K ( x ) K Recall u ′ c ′ = − K K ∇ ¯ c , write G = ∇ ¯ c : K u ′ c ′ = K K xx G x + K K xy G y K K v ′ c ′ = K K yx G x + K K yy G y K K ⇒ four unknowns and two equations. K is property of flow, use two independent tracers a and b forced K K a , Λ = ∇ ¯ by different large-scale gradients: Γ = ∇ ¯ b , u ′ a ′ = K K xx Γ x + K K xy Γ y K K v ′ a ′ = K K yx Γ x + K K yy Γ y K K u ′ b ′ = K K xx Λ x + K K xy Λ y K K v ′ b ′ = K K yx Λ x + K K yy Λ y K K ⇒ four unknowns and four equations.

Tracer a and b a b 1.0 1.0 0.8 0.5 0.6 0.4 0.0 0.2 −0.5 0.0 0 5 10 15 20 30 35 40 45 50 longitude ( x ) latitude ( y )

Measured K K ( x ) K K xx K xy K K K K K yx K yy K K K K

Transport Theory Isotropic local mixing length estimate u ′ 2 + v ′ 2 � V T = L mix = V T / |∇ V T | K xx = K K yy = cV T L mix K K K Shear dispersion in x + Mixing length in y K yy = cV T L jet ( L jet ≈ jet width ) K K K xx = U 2 jet L 2 jet ( U jet ≈ jet speed ) K K K yy K K K. S. Smith, J. Fluid Mech. 544 , 133 (2005)

Isotropic Mixing Length Estimate K xx ( or K K yy ) K K K

Summary K ( x ) in an Study the eddy diffusivity tensor K K inhomogeneous system (the double-gyre configuration) K K ( x ) from high resolution simulation data Measure K Make theoretical estimation on K K K ( x ) in some regions of the domain using mixing length theory Work in progress: K ( x ) improve measurement of K K K ( x ) in use more sophisticated theory to predict K K the whole domain