Effective diffusivities in a two-layer, isopycnal, wind-driven basin - PowerPoint PPT Presentation

Effective diffusivities in a two-layer, isopycnal, wind-driven basin model Yue-Kin Tsang Shafer Smith Center for Atmosphere Ocean Science Courant Institute for Mathematical Sciences New York University

Question(s) • To what degree can baroclinic eddy transport be described in terms of a local eddy closure? • Is there a scale separation between eddy and mean? • Is the small-scale, local eddy flux isotropic or anisotropic? Extremely preliminary results on these questions.... Define eddy relative to low-pass mean, but approximate this via running average in time and space: � t + τ/ 2 � x + ℓ/ 2 � y + ℓ/ 2 1 t − τ/ 2 dt ′ x − ℓ/ 2 dx ′ y − ℓ/ 2 dy ′ f ( x ′ , y ′ , t ′ ) ¯ f ( x, y, t ) = τℓ 2

Large-scale equation Assume advection of passive tracer in one isopycnal layer by quasi- oceanic velocity field: c t + ∇· ( u c ) = κ ∇ 2 c + S Apply our average: c ) = κ ∇ 2 ¯ c + ¯ ¯ c t + ∇· ( F + ¯ u ¯ S where F = u ′ c ′ and primed quantities are deviations from the mean, c ′ = c − ¯ u ′ = u − ¯ c ′ ≃ u ′ ≃ 0 . c, u , such that A perfect numerical model would yield ¯ c subsampled on discrete points separated by distances ℓ and times τ .

The effective diffusivity: A review Assuming that the eddy flux is a linear function of the large-scale mean gradient, we write it as K xx K xy � � F = u ′ c ′ = − K K ∇ ¯ K c, where K K = K = K K K ( x , t ) K yx K yy Can decompose the effective diffusivity into symmetric and antisym- K s + K K a , where metric parts as K K K = K K K � K xx � � � α 0 − γ K s = K a = K − κ I I and K K K I K yy α γ 0 with α = ( K xy + K yx ) / 2 and γ = ( K yx − K xy ) / 2. Eddy transport velocity is u ⋆ = ∇ × ( − ˆ z γ ) = ˆ z × ∇ γ and so averaged flow is c t + ∇· ( u † ¯ K s ∇ ¯ c ) + ¯ ¯ c ) = ∇· ( K K S where u † = u ⋆ + ¯ u

Objectives K is a property of the flow K K 1. Measure K K K directly in an high-resolution ocean model 2. Develop theory to predict K K K from local ocean parameters

Measuring K K K On cell: u ′ c ′ = − K K K Γ (where K K K ≈ constant on cell), or u ′ c ′ = K K xx Γ x + K K xy Γ y K K v ′ c ′ = K K yx Γ x + K K yy Γ y K K ⇒ four unknowns and two equations. Since K K is property of flow, use two independent tracers a and b (forced K by different large-scale gradients ), with Λ ≈ ∇ ¯ Γ ≈ ∇ ¯ a, b So that: 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 K yx Λ x + K K yy Λ y v ′ b ′ = K K K ⇒ four unknowns and four equations.

Model/Simulation Two-layer, adiabatic isopycnal PE model (HIM) in 22 ◦ × 20 ◦ basin • Sinusoidal wind-forcing (double-gyre) and small linear bottom drag • Biharmonic Smagorinsky viscosity plus low-level biharmonic constant background. • 2 passive scalars in top layer, one forced with meridional gradient the other with zonal gradient (** need improved method for gradients) • Current resolution: 1 / 20 degree, but will go higher. • Running mean in space and time calculated using 100km × 100km × 10 day cells, and eddy fluxes of each tracer calculated on each cell (** need bigger cells in ℓ and t )

Snapshot of the flow u ′ v ′ Tracer b

PDF of u ′ and v ′ in cells (Regions far from the jet are nearly isotropic) 0 0 0 0 0 0 10 10 10 10 10 10 −2 −2 −2 10 10 10 −5 −5 −4 −4 −4 −5 10 10 10 10 10 10 −0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2 −0.1 0 0.1 −0.1 0 0.1 −0.2 0 0.2 0 0 0 0 0 0 10 10 10 10 10 10 −2 −2 10 10 −5 −5 −4 −4 −5 −5 10 10 10 10 10 10 −0.2 0 0.2 −0.5 0 0.5 −0.2 0 0.2 −0.1 0 0.1 −0.5 0 0.5 −1 0 1 0 0 0 0 0 0 10 10 10 10 10 10 −2 10 −5 −5 −4 −5 −5 −5 10 10 10 10 10 10 −1 0 1 −0.2 0 0.2 −0.1 0 0.1 −1 0 1 −1 0 1 −1 0 1 0 0 0 0 0 0 10 10 10 10 10 10 −2 10 −5 −4 −5 −5 −5 −5 10 10 10 10 10 10 −0.2 0 0.2 −0.1 0 0.1 −0.5 0 0.5 −1 0 1 −0.5 0 0.5 −0.2 0 0.2 0 0 0 0 0 0 10 10 10 10 10 10 −2 −2 −2 −2 10 10 10 10 −4 −5 −5 −4 −4 −4 10 10 10 10 10 10 −0.1 0 0.1 −0.2 0 0.2 −0.2 0 0.2 −0.1 0 0.1 −0.05 0 0.05 −0.05 0 0.05

Energy spectrum in cells (gives info about eddy energy and scale in each cell: mixing length and mixing velocity) 0 0 −4 0 0 −2 10 10 10 10 10 10 −5 −5 −6 −5 −5 −4 10 10 10 10 10 10 −10 −10 −8 −10 −10 −6 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −10 −10 −10 −10 −10 −10 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −10 −10 −10 −10 −10 −10 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −10 −10 −10 −10 −10 −10 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10 −5 0 0 −5 −5 −5 10 10 10 10 10 10 −5 −5 −10 10 10 10 −10 −10 −10 −10 −10 −15 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10

Zonal and meridional spectra in each cell (second measure of anisotropy on each cell) 0 0 −5 0 0 0 10 10 10 10 10 10 −5 −5 −10 −10 −5 10 10 10 10 10 −10 −10 −10 −20 −20 −10 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −10 −10 −10 −10 −10 −10 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10 0 0 0 0 0 0 10 10 10 10 10 10 −5 −5 −5 −5 −5 −5 10 10 10 10 10 10 −10 −10 −10 −10 −10 −10 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10 0 0 0 0 −5 −5 10 10 10 10 10 10 −5 −10 −5 −5 −10 −10 10 10 10 10 10 10 −10 −20 −10 −10 −15 −15 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10 −5 0 −5 −5 −5 −5 10 10 10 10 10 10 −10 −10 −10 −10 −10 10 10 10 10 10 −15 −20 −15 −10 −15 −15 10 10 10 10 10 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 0 10 1 10 2 2 2 2 2 2 10 10 10 10 10 10

Predicting K K K : Small-scale ’cell’ problem Consider a region (cell) with linear dimension ℓ centered at x i and a period of time τ centered at t i such that 1. ℓ is much smaller than the length scale over which ∇ ¯ c varies, i.e. � ℓ ≪ |∇ ¯ c | � � |∇ 2 ¯ � c | � x i 2. τ is much smaller than the time scale over which ¯ c varies Then in such a cell we have c ( x , t ) ≈ Γ · x + c ′ ( x , t ) , Γ ≈ ∇ ¯ (linear approximation) c and ∂c ′ ∂t + ∇· ( u c ′ ) = κ ∇ 2 c ′ − u · Γ − ¯ c ∇· u Use analytical solutions to simplified cell problem, based on general set of flow types from model, to predict K K ... K

Plans • Increase horizontal resolution • Change cell size • Calculate diffusivity and compare to local theories • Develop suite of cell problem solutions to test • Increase vertical resolution and diagnose eddy PV flux directly...