The Long and Short of Global Modelling of Ocean Tides

THE LONG AND SHORT OF GLOBAL MODELLING OF OCEAN TIDES

Stephen D. Grifﬁths Department of Applied Maths, University of Leeds, U.K. THE BAROTROPIC LUNAR TIDE (M2) sea-surface height (m)

Tidal amplitudes (m), generated from the TPXO6.2 solution www.coas.oregonstate.edu/research/po/research/tide/global.html INTERNAL TIDES Bottom pressure Ocean depth perturbation

forcing Surface (barotropic) tide Internal tide drag Aim: to produce an efﬁcient numerical model of ocean tides, using datasets for astronomical forcing, bathymetry and stratiﬁcation, but with no assimilation of observed data.

1. PHYSICS Challenge: what are the important physical processes?

2. MATHEMATICAL MODELLING Flow of a thin layer of stratified fluid, with a free surface, flowing over complex topography. Challenge: develop equations in a convenient form.

3. NUMERICS Numerical implementation of equations of motion. Challenge: solve the equations (i) in a reasonable time, and (ii) without running out of memory. OUTLINE 1. PHYSICAL BACKGROUND Tidal constituents; relevant physical processes.

2. NUMERICAL MODELLING STRATEGIES Shallow-water models; layered models; OGCMs. An alternative: (i) modal decomposition, (ii) freq. domain. 3. GLOBAL MODELLING OF SURFACE TIDES A fast iterative numerical scheme using sparse inversion.

4. GLOBAL MODELLING OF INTERNAL TIDES Low-order internal waves forced by observed tides. Global estimates of internal tide conversion.

5. SUMMARY & ONGOING WORK Fully coupled surface/internal tide modelling. 1. PHYSICAL BACKGROUND

Mass M (kg) 7.35 1022 1.98 1030 ⇥ 8 ⇥ 11 Distance D (m) 3.84 10 1.5 10 gravitational 2 2 ⇥ 5 ⇥ 3 GM/D (m s )410 6 10 acceleration 3 2 ⇥ 7 ⇥ 7 GMre/D (m s )5.5 10 2.5 10 ⇥ ⇥ tidal acceleration radius of Lunar tide (M2), Earth Solar tide (S2), period 12.42h period 12h GEOMETRY OF THE SUN, MOON, AND EARTH

23.5o Moon Sun 5.1o Earth ecliptic

There is tidal forcing at a range of frequencies, with the largest components being close to diurnal or semi-diurnal. TIDAL CONSTITUENTS

4 Measured 0 (Liverpool) −4 4

0 M2, 12.42h,

−4 amp 3.04m 4

0 S2, 12.00h,

−4 amp 0.98m 4 K1, 23.93h, 0 amp 0.12m −4 Sea-surface height (m) 4 O1, 25.82h, 0 amp 0.12m −4 0 2 4 6 8 10 12 14 (based on data from day of month (July 2008) www.pol.ac.uk/ntslf) OBSERVED TIDAL AMPLITUDES (m) 3 1 M2 1.5 K1 0.5

0.25

0.5

312 PJ 0 50 PJ 0 11 1 S2 0.5 O1 0.5

0.25 0.25

50 PJ 00 25 PJ 0 Tidal amplitudes (m), generated from the TPXO6.2 solution www.coas.oregonstate.edu/research/po/research/tide/global.html SELF-ATTRACTION (AND LOADING) The height variations of the ocean induce a time-varying gravitational potential given by 2˜ = 4⌅G⇧(r r )⇥(⇤, ⌃, t) ⇥ SA e Using an expansion in spherical harmonics, one can write the solution as (e.g., Lamb 1932)

m m im n ˜ n (t)Pn (sin ⇥)e (r/re) r < re SA = 4⇤G⌅re (n+1) 2n + 1 ⇥ (r/re) r > re m n ⇥⇥ We need SA(, ⇥, t) = ˜ SA(r = re) to evaluate horizontal gradients at the Earth’s surface. (see Hendershott 1972 & Farrell 1973, also for discussion of Earth loading) TURBULENT BOTTOM DRAG G.I.Taylor (1919) recognized that a turbulent bottom boundary layer would lead to signiﬁcant drag. He suggested D = ⇢c u u d| | with cd = 0.0025 . Taking

1 2 u 0.25 m s D 0.2 N m . | | ⇥ | | Over all shelf areas, the implied globally integrated energy extracted from barotropic tide is force (per unit area) velocity fractional area of ocean ⇥ ⇥ 2 1 13 2 12 0.2 N m 0.25 m s 2.7 10 m 10 W = 1 TW ⇥ The implied damping timescale is 300 PJ / 1 TW ≈ 3 days. (A SIMPLE VIEW OF) INTERNAL TIDES short wavelength: 10-100km small surface signature: 2-3cm

warm

mixing in internal tide beams large internal displacements: up to 150m density stratiﬁcation

sea-ﬂoor cold THE INTERNAL TIDE AT HAWAII

Figure 2 from Rudnick et al. (Nature, 2003) SUMMARY: THE TIDAL SYSTEM

3.5 TW total=3.5 TW (incl. 2.44 TW for drag forcing lunar recession) Barotropic tide energy Bottom drag forcing drag boundary Vertical layer mixing in energy Internal tide turbulence deep ocean? (circulation, See: Wunsch (2001), Nature, pp.743-744; climate) or Munk & Wunsch (1998), DSR, pp.1977-2010. 2. NUMERICAL MODELLING STRATEGIES Inputs: astronomical forcing, bathymetry, stratiﬁcation, elasticity of Earth (sea-ice, atmospheric forcing?).

1 1. Free-surface shallow-water dynamics c = gH 200 m s ⇡ 2. Global model with ∆x≲1/4 deg≈25km,⇣ so p∆t≲20-100s. ⌘ 3. Account for turbulent boundary layer drag: need good resolution of marginal seas. Parameterize: D = ⇢cd u u 4. Self-attraction+loading: spherical harmonic transform.| | 5. Account for internal wave generation (∆x≪10km?); resolve topography and thin wave beams? 6. Apart from (3), linear solutions may be fine, even for internal tide generation (e.g., Di Lorenzo et al., JPO 2006), but not development and mixing. 7. Efficiency desirable: many different scenarios to test. (SINGLE-LAYER) SHALLOW-WATER MODELS e.g., Jayne and St. Laurent (GRL 2001), Arbic et al. (DSR 2004), Egbert et al. (JGR 2004), Uehara et al. (JGR 2006), Griffiths and Peltier (J.Clim. 2009), Green et al. (GRL, 2009). 1. Parameterize internal tide drag. 2. Start from rest; time step until equilibrium (20 days). 3. Perform a spectral analysis (10-200 days). 4. Tune the internal tide drag coefficients! 5. Resolution: 1/2-1/12th deg. PROBLEMS 1. Self-attraction and loading: approximate treatment. 2. Internal tide drag: non-local process with a non-trivial dependence upon ocean depth, stratification and topographic lengthscales in even the simplest analytical models (e.g., Baines 1973, 1982; Bell 1975; Llewellyn Smith and Young, 2002, 2003). It is frequency dependent too (free waves only when ! < f )! | | | | MULTI-LAYER SHALLOW-WATER MODELS

Arbic et al. (DSR, 2004), Simmons et al. (DSR, 2004), Simmons (OM, 2008). 2-16 layers; resolution 1/2-1/8th deg. Explicit modelling of internal tides, albeit in a simpliﬁed way. Slow evolution of internal tides and fast evolution of surface tides. OCEAN CIRCULATION MODELS Arbic et al. (OM, 2010): HYCOM, 1/12.5 deg, 32 layers. Müller et al. (GRL, 2012): MPI-OM, 1/10 deg, 40 layers.

Fig 7(a) from Arbic et al. (2010). Sea-surface perturbation: model (red), altimetry (blue). degrees

1. Explicit modelling of internal tide generation, sometimes along with additional small-scale parameterized drag. 2. Interaction with ocean general circulation, eddies, sea-ice, etc. 3. Again, no spherical harmonic transforms for self-attraction and loading; a rather simpliﬁed treatment. LINEAR INTERNAL WAVELENGTHS

wavelength(km) at M2 frequency

ocean depth (m) AN ALTERNATIVE APPROACH @u @p @ @w + f u = p + F, = , + N 2(z)w =0, u + =0 @t ⇥ r @z @t r · @z @⇣ w = u H at z = H(x, y); p = g⇣ and w = at z =0. · r @t full b.c. at sea-ﬂoor linearised b.c.s at free surface z = ⇣

Linear waves p(x, z, t)=A(x ct)Z(z) with constant H and f =0: Z0 0 Z 2 + =0,Z0( H)=0,N(0)Z(0) + gZ0(0) = 0. N 2(z) c2 ✓ ◆ Single barotropic mode Inﬁnite set of internal modes

c gH cm NH/m⇡,m 1 0 ⇠ Sturm-Liouville form allows modal expansion of ﬂow in z: u = Un(x, y, t)Zn(z, H(x, y)), p = Pn(x, y, t)Zn(z, H(x, y)). n=0 n=0 MODAL SHALLOW-WATER EQUATIONS (cf. Grifﬁths & Grimshaw, JPO 2007) Barotropic mode: forcing & bottom drag @U D 0 + f U = P IT + F internal wave drag @t ⇥ 0 r 0 ⇢H 0 1 1 @P0 + (HU )=0 DIT = ⇢ H Pn 0 r g @t r · n=1 = H Xbaroclinic pressure r ⇥ at z=-H Internal waves: M @Um 1 barotropic + f Um = Pm H mnPn @t ⇥ r r I forcing n=1 M X @Pm 2 2 1 2 + c U = (c )0U H H c U @t r · m m m 0 · r r · Imn n n n=1 X Here, (c (H),c (H)) and c (H) are horizontally varying. Imn m n m LOW-ORDER 2-DIMENSIONAL SOLUTIONS 0

−0.2 For large-scale

−0.4 topography, most R of the energy ﬂux z / h −0.6 s = 0.8 s = 0.8 s = 0.8 is in low modes m = 0,1 m = 0,1,2 m = 0,1,2,3,4 −0.8 M = 1 M =M=2 2 M =M=4 4 (cf., e.g., Echeverri et al. Q = 0.04 ω h L Q = 0.04 ω h L Q = 0.04 ω h L (2009, JFM)) −1 R s R s R s 0

−0.2

−0.4 We truncate to R four internal z / h −0.6 s = 0.8 s = 0.8 s = 0.8 wave modes. m = 0,1,2,..,8 m = 0,1,2,..,16 m = 0,1,2,..,32 −0.8 M = 8 MM=16 = 16 MM=32 = 32 Q = 0.04 ω h L Q = 0.04 ω h L Q = 0.04 ω h L −1 R s R s R s −1 0 1 2 −1 0 1 2 −1 0 1 2 (x−x ) / L from(x −Grifﬁthsx ) / L & Grimshaw (2007,(x−x ) / L JPO) L s L s L s AN ALTERNATIVE APPROACH 1. With the modal decomposition, the 3D (+time) problem becomes 2D (+time), with coupling between modes. 2. Exploit near-linearity of (i) surface tides, and (ii) internal tide generation. Thus seek solutions at speciﬁed tidal frequencies. (cf. Egbert & Erofeeva (JAOT, 2002), Lyard et al. (Oc. Dyn. 2006)).

Then, @ / @ t i ! , and obtain 2D elliptic PDEs on the sphere, with coupling! between modes. Trade the long runs of time- stepping for memory intensive (sparse) inversions. But what about nonlinearity (drag) and dense operations (SAL)?

To follow: • Barotropic tide solutions, with parametrized internal tide drag. • Internal tide solutions, with a prescribed barotropic tide. • Fully coupled barotropic/internal tides. 3. GLOBAL MODELLING OF BAROTROPIC TIDES

J J (j) i! t (j) i! t Writing U =Re Uˆ e j and P =Re Pˆ e j , 0 0 0 1 0 0 0 1 j=1 j=1 X X @ A forcing @param. drag A sparse and linear terms (nonlinear bottom/ITD)

(j) i! Uˆ + f Uˆ (j) + Pˆ(j) = ˆ(j) ˆ(j) (Pˆ(j)) Dˆ (j) j 0 ⇥ 0 r 0 r r SAL 0 (j) (j) i!jPˆ /g + HUˆ =0 0 r · 0 self attraction & loading: ⇣ ⌘ linear & dense (spherical harmonic transform)

T (j) 1 cd U0 U0 ˆ i!j t where D0 =2 lim e | | dt T 2T T H !1 Z AN ITERATIVE NUMERICAL SOLUTION T ˆ (j) ˆ (j) ˆ(j) For each component write aˆ = U0 , V0 , P0 , giving (L i!I) aˆ = bˆ + ˆs⇣(aˆ) dˆ (aˆ) ⌘ linear dynamics Nonlinear friction 0 f g @ re cos ✓ @ L = f 0 g @ 0 re @✓ 1 forcing self-attraction 1 @ H 1 @ cos ✓H 0 B re cos ✓ @ re cos ✓ @✓ C @ A aˆn+1 + aˆn Solve iteratively, averaging aˆn+1 at each step: ! 2 (L i!I + D S ) aˆ = bˆ +[ˆs(aˆ ) S aˆ ] dˆ (aˆ ) D a n n n+1 n n n n n n sparse approximants to SAL andh nonlinear dragi

• Regular square C-grid, with rotated pole through Greenland/Antarctica. • 2nd-order energy conserving ﬁnite differences. • At 1/4 degree resolution, there are 680,000 grid cells.... • ... and the matrix L is 2million x 2million with 10 million entries. CONVERGENCE

1 10 m2 k1 s2 n o1 0 2 10 / p1 n2 k2 |

1 q1 − n h

− −1 n 10 max | h 5cm −2 10 convergence criterion

−3 10 1 2 3 4 5 6 7 8 9 iteration number n RESULTS: M2 AMPLITUDE AND PHASE 3 3

1.5 1.5

0.75

0.5

0.3

0 0 Model Data (TPXO6.2) 2π 2π

3π/2 3π/2

π π

π/2 π/2

0 0 RESULTS: K1 AMPLITUDE AND PHASE 1 1

0.5 0.5

0.25 0.25

0.1

0 0 Model Data (TPXO6.2) 2π 2π

3π/2 3π/2

π π

π/2 π/2

0 0 A QUANTITATIVE COMPARISON

Energy Dissipation

TIDE KE (PJ) PE (PJ) DBL (TW) DIT (TW) D (TW) M2 model 201 172 1.530 1.392 2.923 M2 obs 178 134 < 1.649 > 0.782 2.435 K1 model 36.8 19.8 0.199 0.134 0.333 K1 obs 31.4 18.5 < 0.304 > 0.039 0.343

Remember: these are solutions with a parametrized internal tide drag. One can get almost any result possible by tuning the internal tide drag scheme... 4. GLOBAL INTERNAL TIDE MODELLING

Barotropic mode: forcing & bottom drag @U D 0 + f U = P IT + F internal wave drag @t ⇥ 0 r 0 ⇢H 0 1 1 @P0 + (HU )=0 DIT = ⇢ H Pn 0 r g @t r · n=1 = H Xbaroclinic pressure r ⇥ at z=-H Internal waves: M @Um 1 barotropic + f Um = Pm H mnPn @t ⇥ r r I forcing n=1 M X @Pm 2 2 1 2 + c U = (c )0U H H c U @t r · m m m 0 · r r · Imn n n n=1 X Here, (c (H),c (H)) and c (H) are horizontally varying. Imn m n m LOCAL MODELLING LOCAL 3-D MODELLING: INTERNAL TIDES •Calculate the time-harmonic response for internal i!t wave modes Um(x,t)=Re Uˆ m(x)e , m=1,2,3,4, via mode-coupling ⇣ ⌘ i!Uˆ m + f Uˆ m = Pˆm terms neglected for ⇥ r low-order modelling! 2 2 i!Pˆ + c Uˆ = (c )0Uˆ H m r · m m m 0 · r ⇣ ⌘ •Calculate average N(z) on domain from World Ocean Atlas, and then c m ( H ( x )) from eigenvalue problem. •Equations are discretized to a 2D sparse system: (L i!I) aˆ = bˆ barotropic forcing linear dynamics unknowns u, v, p •Solve with sponge layers (rather than radiation). M2: NOVA SCOTIA

Ocean depth (m) Internal wave pressure N /m2 Mode 1 Mode 2

M2 Internal Mode 3 Mode 4 wave pressure N/m2 HAWAII

M2 (19.4 GW) Bottom pressure (Pa)

K1 (1.7 GW) Bottom pressure (Pa) M2 (4 GW) Bottom pressure (Pa)

Aleutian Islands M2 (4 GW) Bottom pressure (Pa)

K1 (39 GW) Bottom pressure (Pa) M2 CONVERSION:EGBERT AND RAY: TIDAL TPXO.4A ENERGY DISSIPATION vs I.T. MODEL 22,485

-20

-40

-60 0 50 100 150 200 250 300 350 Figure2. Shallowseas and deep ocean areas used for integrated dissipation computations. Areas outlined with solidfromlines, numbered JGR,1-28, 2001 includeall of theEgbert+Ray shallowseas where significant Modeldissipation due to bottom boundarylayer drag would be expected. In particular,essentially all of thedissipation in theprior model (Plate2a) occurswithin these areas. Areas outlined with(GW) dashedlines and (GW)labeled A-I are deep-water areaswhich showenhanced dissipation in the T/P estimatesof Plate2. Not enough A: Micronesia/Melanesia 100 72 modes for rough whichsimulate C:the Mid-Atlantic interactionof thebarotropic Ridgetide with to- -V. 103U/iro agreeswith the 47 tidal elevationtopography;•' to within about pographicfeatures F: Polynesiapresent in the fine grid but not resolved 1 cm 31 overmost of the open 39ocean [Ray, see2001 Zilberman]. by the coarsergrid. The pairingof red andblue spotsaround The dissipationmap from TPXO.4aet al.has (2009), the cleanest JPOap- this unresolvedH:topography Hawaiiimplies that theseresiduals on pearance, 18while the map 19 from the purely empirical DW95 averagedo no net work aroundany particularfeature. We solutionis noisiest,with significantareas of negativedissi- demonstratethis more explicitly below by computingarea pation throughoutthe ocean. However, all three maps have integralsof dissipationfor the prior solution. Becausethe many featuresin common. The areasof intensedissipation real oceanhas topographythat will not be resolvedin the expecteddue to bottom boundarylayer drag in the shallow dynamical equationsused to estimatecurrents, we should seas (e.g., Patagonian Shelf, Yellow Sea, northwest Aus- expectsimilar sortsof noise(and areasof negativedissipa- tralianShelf, European Shelf) are clearlyevident in all cases tion) in all of our estimatesof D. (compareto the prior solution of Plate 2a). The areas of The remaining three panels of Plate 2 show dissipation enhancedopen oceandissipation discussed in E-R are also mapsbased on threetidal solutionsrepresentative of the dif- seenin mapsfor all threeT/P-constrained solutions (but not ferent approachesused to estimatetides from T/P data: for- for the prior solution). For example, dissipationis clearly mal assimilationmethods, empirical corrections to a hydro- enhancedin the Pacificover the Hawaiian Ridge, the Tu- dynamicmodel, and purely empirical solutions. The first amotu archipelago,and over the back arc islandchains ex- two of thesemaps (Plates2b and 2c) are the TPXO.4a and tendingfrom Japansouthward to New Zealand. The west- GOT99hf estimatespresented in E-R. The third approach, ern IndianOcean around the MascereneRidge and south of a purelyempirical solution with no explicitreliance on any Madagascaralso exhibitsenhanced dissipation in all three dynamicalassumptions, is representedby the DW95 solu- of themaps. The Mid-Atlantic Ridge shows up mostclearly tion of Desai and Wahr[ 1995] (Plate 2d). For the GOT99hf in the assimilation estimate but is evident also in the other and DW95 solutions,volume transports were computedus- two estimates. In all three cases there is substantial dissi- ing the least squaresapproach of section4.2, with the lin- pation throughoutmuch of the North Atlantic. All of these ear friction parameterof (22) set to a relatively high value areaswhere dissipation is consistentlyenhanced are charac- (r = 0.03), and the weight wc of (25) chosenso that terizedby significantbathymetric variation, generally with

Plate 1. Termsin the time averagedenergy balance equation (12), derivedfrom the assimilationsolution TPXO.4a. In Plates la throughld the work termsare subdividedas in (16)-(19). (a) The work done on the oceanby the large-scalegravitational potential, and (b) the work doneby potentialterms associated with loadingand self-attraction.(c) and (d) Work done by the moving bottomon the ocean,associated with thebody tide •'b,and load tide •'t,respectively. (e) The sumof thefour work termsplotted in Platesla throughld, and(f) the divergenceof the energyflux V ßP. Note that Platesle and I f nearlycancel; the differenceis usedto estimatethe oceanictidal energydissipation rate. M2: BAROCLINIC CONVERSION 1000

100

1 2

−1 mW m

−10

−100

−1000 I.T. MODEL: 0.62 TW = 0.41 TW (open ocean) + 0.21 TW (coastal) TPXO.5: 2.43 TW = 0.78 TW (open ocean) + 1.65 TW (coastal) K1: BAROCLINIC CONVERSION 1000

18 GW 100 30 GW 10 39 GW

1 2

−1 mW m

55 GW −10

−100

−1000 I.T. MODEL: 0.23 TW = 0.10 TW (open ocean) + 0.13 TW (coastal) TPXO.5: 0.34 TW = 0.04 TW (open ocean) + 0.30 TW (coastal) 5. SUMMARY AND ONGOING WORK •Barotropic tides can be modelled using quasi-linear shallow water equations, and seeking solutions at ﬁxed frequencies. •Global multi-constituent solutions can be found using sparse matrix inversion, iterated to account for parameterized nonlinear bottom drag & self-attraction and loading. •This is much faster than traditional time-stepping: about 10min per constituent, vs 10h per constituent (on a desktop machine).

•Linear 3D hydrostatic ﬂow over large amplitude topography. •Use a vertical modal decomposition to target internal waves. •Work in the frequency domain; solve large sparse systems of 2D PDEs by direction inversion (fast, memory intensive). •Find local solutions in 112 patches covering the entire globe. •Use internal modes 1,2,3,4; solve at 1/40th degree. •See wave generation by slopes and ridges, and some rough topography, roughly consistent with observational estimates. • Our internal tide model is sufﬁciently realistic (and fast) to be used as a super-parameterization of internal tide drag in frequency-based global models of the barotropic tide.

Calculate Calculate surface tide, internal tide drag globally, with and add to bottom friction barotropic equations

Calculate internal tide (modal decomposition, 100+ subdomains) A CONVERGING SOLUTION

1 10 CONVERGENCE! A single constituent solution (M2) m2

Switch on Fully coupled internal tides 0 iterations 10

| (5hr each, in serial) 1 − n h − n max | h −1 Barotropic-only 10 iterations over nonlinear friction and difference in solution self-attraction & loading (2min each) −2 10 2 4 6 8 10 12 14 16 18 iteration number n iteration number 5. SUMMARY AND OUTLOOK •Tidal models need to account for astronomical forcing, realistic bathymetry, bottom drag, internal tide generation, and gravitational self-attraction and Earth loading (SAL). •Modern tidal models based on time-stepping of layered models are computationally expensive (long run times), barely resolve internal tides (even at 1/12th deg), and use approximate SAL. •Working in the frequency domain is much faster, and allows exact implementation of SAL in global models. Coupled with local low- order internal tide models, one obtains a global tidal model with high resolution (1/20-1/40th deg), which can be run on a desktop.

FUTURE WORK (see Vladimir Lapin) •More baroclinic modes, mode coupling, weak nonlinearity, .... •Improve implementations for large elliptic solves. •Parallel implementation of internal tide model. •Targeted resolution in shallow-seas and over topography.