A Meshless Approach to Spectral Wave Modeling Adrean Webb The - PowerPoint PPT Presentation

A Meshless Approach to Spectral Wave Modeling Adrean Webb The University of Tokyo Graduate School of Frontier Sciences August 7, 2017 Natasha Flyer (NCAR), Baylor Fox-Kemper (Brown University) In Collaboration with:

Outline 1. Scientific background 2. Coupled wave component 3. Overview of meshless approach 4. Kinematic prototype 5. Discussion Adrean Webb / 39 2

Scales of interest arbitrary energy scale tides seiches wind-generated waves --------------- trans-tidal waves surges __ ts_t_m_a_m_i_s ___ _ }nfra-gravity waves - - ---- ----"'------- capillary swell wind sea waves 10-3 10-1 10-6 10-5 J0-4 IQ-2 10° frequency (Hz) 10+ 1 3h 15min lOs 24 h 100 s period 0.1 s 1 s Figure: Illustration of wave spectra from di ff erent types of ocean surface waves (Holthuijsen, 2007) . Figure 1.1 Frequencies and periods of the vertical motions of the ocean surface (after Munk, 1950). Adrean Webb / 39 3

Spectral wave model approach The statistical properties on intermediate scales are assumed to vary smoothly with location and the generation, propagation, and evolution of wave spectra are modeled deterministically. Figure: The random sea of each gridded region in (a) is Fourier decomposed in (b). The statistical di ff erences between neighboring gridded regions are assumed to be small enough such that advection and evolution of wave energy can be modeled by a PDE (Holthuijsen, 2007) Adrean Webb / 39 4

Sample spectral model output Figure: Global surface Stokes drift on a 1 deg lat x 1.25 deg lon grid. Adrean Webb / 39 5

Wave action balance equation x , ~ k 2 R 2 ) from The wave action balance equation can be derived on a coupled slab ( ~ the governing linear wave equations and is given by @ t W + r k Ω · r x W � r x Ω · r k W = Sources ~ ~ ~ ~ where o 1 / 2 n h i x , ~ g | ~ | ~ � ~ � • � = k | tanh k | H ( ~ x ) is the intermediate-water dispersion k relation x , ~ = ~ · ~ x , ~ � ~ � � ~ � � ~ � • Ω k + � is the Doppler-shifted dispersion relation k U x k x , ~ � ~ x , ~ � ~ � � � ~ � • W k , t = g S ~ k ; ~ x , t / � is the spectral adiabatic invariant k k • “Sources” encompass the non-kinematic physics of the waves (generation, dissipation, nonlinear interactions, etc.) Adrean Webb / 39 6

Wave action balance equation x , ~ k 2 R 2 ) from The wave action balance equation can be derived on a coupled slab ( ~ the governing linear wave equations and is given by Spatial transport @ t W + r k Ω · r x W � r x Ω · r k W = Sources ~ ~ ~ ~ Spatial changes where o 1 / 2 n h i x , ~ g | ~ | ~ � ~ � • � = k | tanh k | H ( ~ x ) is the intermediate-water dispersion k relation x , ~ = ~ · ~ x , ~ � ~ � � ~ � � ~ � • Ω k + � is the Doppler-shifted dispersion relation k U x k x , ~ � ~ x , ~ � ~ � � � ~ � • W k , t = g S ~ k ; ~ x , t / � is the spectral adiabatic invariant k k • “Sources” encompass the non-kinematic physics of the waves (generation, dissipation, nonlinear interactions, etc.) Adrean Webb / 39 7

Wave action balance equation x , ~ k 2 R 2 ) from The wave action balance equation can be derived on a coupled slab ( ~ the governing linear wave equations and is given by Spectral transport @ t W + r k Ω · r x W � r x Ω · r k W = Sources ~ ~ ~ ~ Spectral changes where o 1 / 2 n h i x , ~ g | ~ | ~ � ~ � • � = k | tanh k | H ( ~ x ) is the intermediate-water dispersion k relation x , ~ = ~ · ~ x , ~ � ~ � � ~ � � ~ � • Ω k + � is the Doppler-shifted dispersion relation k U x k x , ~ � ~ x , ~ � ~ � � � ~ � • W k , t = g S ~ k ; ~ x , t / � is the spectral adiabatic invariant k k • “Sources” encompass the non-kinematic physics of the waves (generation, dissipation, nonlinear interactions, etc.) Adrean Webb / 39 8

Outline 1. Scientific background 2. Coupled wave component 3. Overview of meshless approach 4. Kinematic prototype 5. Discussion Adrean Webb / 39 9

Scientific Motivation There is a persistent, shallow mixed layer bias in the Southern Ocean in global climate models (GCMs): Langmuir turbulence missing?? Figure: Satellite observations of Langmuir Figure: Mixed layer bias is reduced in NCAR CESM model runs turbulence after the Deepwater Horizon oil spill (Li, Fox-Kemper, Breivik, & Webb, 2017). (Quickbird) Other studies: Belcher et al. 2012; D’Asaro et al., 2014; Fan & Griffies, 2014 Adrean Webb / 39 10

NCAR CESM now includes a prognostic wave field Modified NOAA WAVEWATCH III has been coupled to NCAR CESM (Craig, Li, Vertenstein, Webb) Third-generation model: Drawbacks for climate use: • Creates and evolves wave spectra in a • Computational expensive coupled spatial-spectral domain • 4D problem in time with 6-50 x 10^6 unknowns • Uses structured grids (lat-lon, polar) and nonlinear source terms • Includes extensive physics and • Spatial and spectral singularities at the poles parameterizations • Difficult to model polar-ice-free climate scenarios 25 freq x 24 dir 3.2 lat x 4 lon (deg) Figure: Sample spatial (SWH) and spectral output for the coupled NCAR CESM wave component. Adrean Webb / 39 11

Simple WAVEWATCH III cost analysis WAVEWATCH III version 3.14 • Let N be the number of grid cells. Then running costs are Cost( N ) = α N 1 . 07 . • Using source terms approximately doubles cost since log 2 [Cost src ] ≈ 1 + log 2 [Cost in ] Cost src ≈ 2 Cost in • Changing grid resolution can have a dramatic e ff ect on speed N global = N x × N y N coarse = ( N x / 3 . 2) × ( N y / 3 . 2) Cost global ≈ 10 Cost coarse Figure: WAVEWATCH III cost versus spatial resolution using same time step and fixed spectral grid (25 f × 24 θ ). Adrean Webb / 39 12

Outline 1. Scientific background 2. Coupled wave component 3. Overview of meshless approach 4. Kinematic prototype 5. Discussion Adrean Webb / 39 13

RBF and RBF-FD methods Radial Basis Functions (RBF): • Uses a meshless node layout • Solves advective problems with spectral accuracy • Allows geometric flexibility and local node refinement • No advective singularities • Smooth solutions require much fewer unknowns compared with other methods RBF-Generated Finite Di ff erences (RBF-FD): • Uses local RBF-generated finite di ff erence formulas to reduce memory count from O ( N 2 ) to O ( N ) • Well-suited for parallelization ( Bollig et al., 2012 ) Figure: Example node layout of a bumpy sphere ( Fuselier & Wright, 2012) and possible solid body rotation over the surface (Flyer & Wright, 2007) Adrean Webb / 39 14

Solving PDEs with the local RBF-FD method Local RBF-FD Method: • Creates a set of FD weights (w) that are exact at point ~ ↵ s for the RBF ( � ) centered at each node in the subset · · · 1 ( L � ) s 1 � 11 � 1 n w 1       . . . . . ... . . . . .       . . . . .  =             · · · 1 ( L � ) sn � n 1 � nn w n      1 · · · 1 0 0 w n +1 where � ji = � ( k ~ ↵ � ~ ↵ i k ) | ~ ↵ = ~ ↵ j • Local weights are combined into a banded sparse matrix D such that LW ( ~ ↵ 1 , t ) · · · W ( ~ ↵ 1 , t )    D 11 D 1 N    . . . .  ⇡ ... . . . .      . . . . LW ( ~ ↵ N , t ) W ( ~ ↵ N , t ) D N 1 · · · D NN Sample (a) local di ff erenti- Figure: ation weights and (b) global banded matrix (Flyer et al., 2012) Adrean Webb / 39 15

Stabilizing the RBF-FD method with hyperviscosity Hyperviscosity Filter: • A high-order Laplacian operator is used to stabilize the RBF-FD method ∂ t W = − D W + γ ∆ p W • The artificial di ff usion coe ffi cient is typically in the range of 10 − 20 to 10 − 45 Figure: Sample eigenvalue spectrum with the RK4 stability domain (solid line) (a) without hyperviscosity and (b) with a ∆ 4 -type hyperviscosity added. (Flyer et. al., 2012) Adrean Webb / 39 16

Outline 1. Scientific background 2. Coupled wave component 3. Overview of meshless approach 4. Kinematic prototype 5. Discussion Adrean Webb / 39 17

Coupled model domains Spatial: Directional-Frequency: WAVEWATCH III: RBF-FD WAVE: Adrean Webb / 39 18

RBF-FD-W model (full-4D) RBF-FD-W Details: • � ( r ) = e � ( ε r ) 2 for " , r 2 R ⇤ + x , ~ • ~ ↵ = ( ~ k ) = ( x , y , z , k x , k y , k z ) m x , ~ ↵ = ( ~ f ) = ( x , y , z , ✓ x , ✓ y , f ) ~ • Polar ice caps for latitudes above and below ± 75 � • For constant depth with no currents, the wave action balance Figure: Sample coupled spatial and directional-frequency do- equation is given below: main c g ( k z ) ˆ x 2 + y 2 � n� ⇣ � zk 2 ⌘ o � � � � � � ∂ t W + ∂ x + ∂ y + k y ∂ z + ∂ kx + ∂ ky � yk x � xzk y xk x � yzk y zk x k y W x x 2 + y 2 p = Sources Adrean Webb / 39 19

RBF-FD-W stencil selection (full-3D) Convergence rates are determined by both the total number of nodes and local stencil size (a) Spatial stencil (b) Directional stencil Figure: Sample spatial and directional stencils for spatial node (1 , 0 , 0) and dominant direction π / 6. The combined stencil is 17 ~ x × 9 ~ k = 153 ~ ↵ . Adrean Webb / 39 20