Large Eddy Simulation of Mixing in Stratified Flows Tamay Özgökmen, Traian Iliescu and Paul Fischer U. Miami, Virginia Tech., and Argonne National Lab. SIAM, Long Beach, March 2011 1

Challenges of Modeling Oceanic Mixing: 0) Almost all oceanic flows are turbulent: * mixing is important in coastal zones, down-welling/up-welling, polar and marginal seas, surface flows, overturning circulation in climate modeling... 1) Modeling via Direct Numerical Simulation (DNS): All turbulent scales are resolved using Navier-Stokes (no “parameterization”), but: * for oceanic flows: Re ~ O(108)-O(1012) because of their large scale * degrees of freedom (range of turbulent features) ~ Re9/4 * number of spatial points needed: O(1018)-O(1027) 2) Modeling via Primitive Equations - OGCMs: * Hydrostatic approximation breaks down much before reaching overturning scales * Typically rich with parameters and parameterizations: algebraic (e.g. KPP) or second-order closures (SOC) are used to represent mixing Questions: * Are OGCMs + SOC accurate enough? * Are there any other modeling avenues?

LES: Exploit the Concept of Turbulent Coherent Structures Lab experiment: Re=4300 Discharge from a ship: Re ~107 from p. 100 of Van Dyke (1982) Large Eddy Simulation (LES): Mixing by the large, energy-containing, anisotropic, geometry-dependent, long-lived eddies is handled through computation, while the effect of small, isotropic, dissipative, short-lived turbulent eddies is modeled analytically.

Model: Non-hydrostatic Spectral Element Model nek5000 (developed by Paul F. Fischer, Argonne National Lab.) Combines geometrical flexibility of finite element method and numerical accuracy of spectral expansion. Virtually free of spurious modes: minimal numerical dissipation and dispersion errors. Exponential convergence: small number of spatial points needed to capture the solution. Excellent scalability on parallel machines (up to P=160k).

What are the basic requirements for a test problem? Must contain the three characteristics of stratified flows: (1) Mixing due to stably-stratified (shear-induced) motions. (2) Mixing due to unstably-stratified (convective) motions. (3) Internal waves. Preferably all of them co-existing simultaneously and interacting… Must be free of any “implicit” factors that can affect mixing, such as touch ups to boundary/initial conditions, forcing, domain geometry, etc. Must be as simple as possible to set up. 9

17th order, ~100 million pts, ~100k CPU hours Lock Exchange Problem: 17th order, ~100 million pts, ~100k CPU hours 10

Preliminary Impressions on the SGS Models: DNS* with 203k pts DNS* with 45k pts, no SGS LES Rational SGS 45k pts LES Smagorinsky SGS-A 45k pts

Quantification of Mixing: before mixing: after mixing: We all know that mixing increases PE, but how to apply to complex computations? Snapshot, PE: pdf sorted, RPE: Use the pdf technique of Tseng and Ferziger (2001, Phys. Fluids) for background/reference PE (RPE); minimum PE a given state can attain. APE = PE – RPE RPE increases only by mixing in enclosed domains (Winters et al., 1995)

Evaluate SGS models using mixing metric RPE and by comparing: DNS: fully (or highly) resolved; bite the bullet for gold standard for computational accuracy DNS*: under-resolved, without SGS LES: under-resolved, with SGS Objective: major computational gain [O(1000) in CPU time] with LES with respect to DNS while matching mixing accuracy of DNS 13

CPU time : 76 h or 1,212 times shorter than DNS Points : 134 times less than DNS

How Does Mixing Change as a Function of Re? Re=3x103 Re=104 Re=3x104

A Strategy to Estimate Mixing at High Re: Step 1) Compute mixing using DNS for feasible range: 1000 < Re < 3x104 Step 2) Evaluate LES using DNS results from 1000 < Re < 3x104 Step 3) Estimate mixing beyond DNS range, at Re=105 and Re=106 using LES

Step 1) Step 2) Step 3) All details are in: Özgökmen, Iliescu and Fischer, 2009a,b, Ocean Modelling.

Submesoscale Ocean Application: Mixed Layer Adjustment * Important for multi-scale energy cascade in the ocean, naval operations, air-sea interaction, biogeochemical transport * Lock-Exchange with Rotation and High Aspect Ratio O(100)

Tracer Release in the Mixed Layer:

Tracer Release Below the Mixed Layer:

Summary: 1) Rational SGS model seem to offer an exciting complement to traditional dynamic EV models in a stratified mixing problem. 2) Little sensitivity of mixing is found on SGS models in the density perturbation equation. 3) More complex and more oceanographically relevant surface MLI problem seems to have distinctly different mixing characteristics, driven by long- lasting turbulent features that are maintained by backward energy cascade due to rotation. Supported by NSF-CMG