THE LONG AND SHORT OFGLOBAL MODELLING OF OCEAN TIDES

Stephen D. GriffithsDepartment of Applied Maths, University of Leeds, U.K.

THE BAROTROPIC LUNAR TIDE (M2)

Tidal amplitudes (m), generated from the TPXO6.2 solutionwww.coas.oregonstate.edu/research/po/research/tide/global.html

sea-

surf

ace

heig

ht (

m)

INTERNAL TIDES

Surface (barotropic) tide Internal tideforcing

drag

Ocean depthBottom pressure

perturbation

1. PHYSICSChallenge: what are the important physical processes?

2. MATHEMATICAL MODELLINGFlow of a thin layer of stratified fluid, with a free surface,

flowing over complex topography. Challenge: develop equations in a convenient form.

3. NUMERICSNumerical implementation of equations of motion.

Challenge: solve the equations (i) in a reasonable time, and (ii) without running out of memory.

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.

OUTLINE

1. PHYSICAL BACKGROUNDTidal constituents; relevant physical processes.

3. GLOBAL MODELLING OF SURFACE TIDESA fast iterative numerical scheme using sparse inversion.

4. GLOBAL MODELLING OF INTERNAL TIDESLow-order internal waves forced by observed tides.

Global estimates of internal tide conversion.

5. SUMMARY & ONGOING WORKFully coupled surface/internal tide modelling.

2. NUMERICAL MODELLING STRATEGIESShallow-water models; layered models; OGCMs.

An alternative: (i) modal decomposition, (ii) freq. domain.

1. PHYSICAL BACKGROUND

Mass M (kg) 7.35⇥ 1022 1.98⇥ 1030

Distance D (m) 3.84⇥ 108 1.5⇥ 1011

GM/D2 (m s�2) 4⇥ 10�5 6⇥ 10�3

GMre/D3 (m s�2) 5.5⇥ 10�7 2.5⇥ 10�7

Solar tide (S2), period 12h

Lunar tide (M2), period 12.42h

gravitational acceleration

tidal acceleration

radius of Earth

ecliptic

Moon23.5o

5.1o

GEOMETRY OF THESUN, MOON, AND EARTH

SunEarth

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

TIDAL CONSTITUENTS

day of month (July 2008)

Sea-

surf

ace

heig

ht (

m)

Measured(Liverpool)

M2, 12.42h,amp 3.04m

S2, 12.00h,amp 0.98m

K1, 23.93h,amp 0.12m

O1, 25.82h,amp 0.12m

(based on data from

www.pol.ac.uk/ntslf)

!4

0

4

!4

0

4

!4

0

4

!4

0

4

0 2 4 6 8 10 12 14

!4

0

4

0

0.25

0.5

1

K1

50 PJ

0

0.25

0.5

1

S2

50 PJ

0

0.5

1

1.5

3

M2

312 PJ

0

0.25

0.5

1

O1

25 PJ

www.coas.oregonstate.edu/research/po/research/tide/global.html

Tidal amplitudes (m), generated from the TPXO6.2 solution

OBSERVED TIDAL AMPLITUDES (m)3 1

110 0

00

SELF-ATTRACTION (AND LOADING)

The height variations of the ocean induce a time-varying gravitational potential given by

We need to evaluate horizontal gradients at the Earth’s surface.

Using an expansion in spherical harmonics, one can write the solution as (e.g., Lamb 1932)

(see Hendershott 1972 & Farrell 1973, also for discussion of Earth loading)

⇥2�SA = 4⌅G⇧�(r � re)⇥(⇤,⌃, t)

�SA = �4⇤G⌅re⇥

m⇥n

�mn (t)Pmn (sin ⇥)eim�

2n+ 1⇥�

(r/re)n r < re(r/re)�(n+1) r > re

�SA(�,⇥, t) = �SA(r = re)

TURBULENT BOTTOM DRAG

D = ⇢cd|u|u

G.I.Taylor (1919) recognized that a turbulent bottom boundary layer would lead to significant drag. He suggested

cd = 0.0025with . Taking

force (per unit area) velocity fractional area of ocean⇥ ⇥

Over all shelf areas, the implied globally integrated energy

extracted from barotropic tide is

|u| � 0.25ms�1 ⇥ |D| � 0.2Nm�2.

0.2Nm�2 � 0.25m s�1 � 2.7� 1013 m2 ⇥ 1012 W = 1TW

The implied damping timescale is 300 PJ / 1 TW ≈ 3 days.

(A SIMPLE VIEW OF) INTERNAL TIDESshort wavelength: 10-100km small surface signature: 2-3cm

mixing in internal

tide beams large internal

displacements: up to 150m

sea-floor

dens

ity s

trat

ifica

tion

warm

cold

THE INTERNAL TIDE AT HAWAII

Figure 2 from Rudnick et al. (Nature, 2003)

SUMMARY: THE TIDAL SYSTEM

Barotropic tide

Internal tideVertical

mixing in deep ocean?

(circulation, climate)

Bottom boundary

layer turbulence

forcingdrag

forcingdrag

energy

energy

drag

3.5 TW(incl. 2.44 TW for lunar recession)

total=3.5 TW

See: Wunsch (2001), Nature, pp.743-744;or Munk & Wunsch (1998), DSR, pp.1977-2010.

2. NUMERICAL MODELLING STRATEGIESInputs: astronomical forcing, bathymetry, stratification,

elasticity of Earth (sea-ice, atmospheric forcing?).

1. Free-surface shallow-water dynamics 2. Global model with ∆x≲1/4 deg≈25km, so ∆t≲20-100s.3. Account for turbulent boundary layer drag: need good resolution of marginal seas. Parameterize: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.

D = ⇢cd|u|u

⇣c =

pgH ⇡ 200m s�1

⌘

(SINGLE-LAYER) SHALLOW-WATER MODELS

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.

PROBLEMS1. 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 )!

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).

|!| < |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 MODELSArbic et al. (OM, 2010): HYCOM, 1/12.5 deg, 32 layers. Müller et al. (GRL, 2012): MPI-OM, 1/10 deg, 40 layers.

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.

degrees

Fig 7(a) from Arbic et al. (2010).

Sea-surface perturbation:model (red),

altimetry (blue).

LINEAR INTERNAL WAVELENGTHS

ocean depth (m)

wav

elen

gth

(km

)

at M2 frequency

AN ALTERNATIVE APPROACH

w = �u ·rH at z = �H(x, y); p = g⇣ and w =@⇣

@tat z = 0.

Single barotropic modec0 �

�gH

Infinite set of internal modes

u =��

n=0

Un(x, y, t)Zn(z,H(x, y)), p =��

n=0

Pn(x, y, t)Zn(z,H(x, y)).

Sturm-Liouville form allows modal expansion of flow in z:

linearised b.c.s at free surfacefull b.c. at sea-floor

cm ⇠ NH/m⇡, m � 1

z = ⇣

Linear waves p(x, z, t) = A(x� ct)Z(z) with constant H and f = 0 :

@u

@t+ f ⇥ u = �rp+ F,

@p

@z= �,

@�

@t+N2(z)w = 0, r · u+

@w

@z= 0

✓Z 0

N2(z)

◆0+

Z

c2= 0, Z 0(�H) = 0, N2(0)Z(0) + gZ 0(0) = 0.

MODAL SHALLOW-WATER EQUATIONS

forcing & bottom drag

internal wave drag

Barotropic mode:

1

g

@P0

@t+r · (HU0) = 0

@U0

@t+ f ⇥U0 = �rP0 �

DIT

⇢H+ F0

DIT = �⇢rH1X

n=1

Pn

= �rH ⇥ baroclinic pressure

at z=-HInternal waves:@Um

@t+ f ⇥Um = �rPm �rH

1X

n=1

ImnPn

@Pm

@t+r ·

�c2mUm

�= �(c2m)0U0 ·rH �rH ·

1X

n=1

Imnc2nUn

Here, Imn(cm(H), cn(H)) and cm(H) are horizontally varying.

(cf. Griffiths & Grimshaw, JPO 2007)

barotropic forcing

M

M

LOW-ORDER 2-DIMENSIONAL SOLUTIONS

−1

−0.8

−0.6

−0.4

−0.2

0

z / h

R

s = 0.8m = 0,1M = 1Q = 0.04 ω hRLs

−1 0 1 2−1

−0.8

−0.6

−0.4

−0.2

0

(x−xL) / Ls

z / h

R

s = 0.8m = 0,1,2,..,8M = 8Q = 0.04 ω hRLs

s = 0.8m = 0,1,2M = 2Q = 0.04 ω hRLs

−1 0 1 2(x−xL) / Ls

s = 0.8m = 0,1,2,..,16M = 16Q = 0.04 ω hRLs

s = 0.8m = 0,1,2,3,4M = 4Q = 0.04 ω hRLs

−1 0 1 2(x−xL) / Ls

s = 0.8m = 0,1,2,..,32M = 32Q = 0.04 ω hRLs

For large-scale topography, most of the energy flux is in low modes

(cf., e.g., Echeverri et al. (2009, JFM))

from Griffiths & Grimshaw (2007, JPO)

We truncate to four internal wave modes.

M=2 M=4

M=16 M=32

AN ALTERNATIVE APPROACH

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)).

1. With the modal decomposition, the 3D (+time) problem becomes 2D (+time), with coupling between modes.

To follow:• Barotropic tide solutions, with parametrized internal tide drag. • Internal tide solutions, with a prescribed barotropic tide. • Fully coupled barotropic/internal tides.

Then, , 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)?

@/@t ! �i!

3. GLOBAL MODELLING OF BAROTROPIC TIDES

sparse and linear termsforcing

where

Writing U0 = Re

0

@JX

j=1

U(j)0 e�i!jt

1

A and P0 = Re

0

@JX

j=1

P (j)0 e�i!jt

1

A ,

�i!jP(j)0 /g +r ·

⇣HU

(j)

0

⌘= 0

D(j)

0 = 2 limT!1

1

2T

Z T

�Tei!jt cd|U0|U0

Hdt

�i!jU(j)

0 + f ⇥ U(j)0 +rP (j)

0 = �r�(j) �r�(j)SAL(P

(j)0 )� D(j)

self attraction & loading: linear & dense

(spherical harmonic transform)

param. drag (nonlinear bottom/ITD)

AN ITERATIVE NUMERICAL SOLUTION

Solve iteratively, averaging

Nonlinear friction

forcing

linear dynamics

(L� i!I) a = b + s (a)� d (a)

self-attraction

(L� i!I + Dn � Sn) an+1 = b + [s(an)� Snan]�hd (an)�Dnan

i

For each component write , giving

L =

0

B@0 �f g

re cos ✓@

@�

f 0

gre

@@✓

1

re cos ✓@

@�H 1

re cos ✓@@✓ cos ✓H 0

1

CA

sparse approximants to SAL and nonlinear drag

• 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.

a =⇣U (j)0 , V (j)

0 , P (j)0

⌘T

an+1 !an+1 + an

2 at each step:

CONVERGENCE

1 2 3 4 5 6 7 8 910

!3

10!2

10!1

100

101

iteration number n

ma

x | h

n!

hn

!1 |

m2k1s2o1p1n2k2q1

5cm convergence

criterion

/ 2�n

RESULTS: M2 AMPLITUDE AND PHASE

Model Data (TPXO6.2)0

0.3

0.75

1.5

3

0

0.5

1

1.5

3

0

!/2

!

3!/2

2!

0

!/2

!

3!/2

2!

RESULTS: K1 AMPLITUDE AND PHASE

Model Data (TPXO6.2)0

0.1

0.25

0.5

1

0

0.25

0.5

1

0

!/2

!

3!/2

2!

0

!/2

!

3!/2

2!

A QUANTITATIVE COMPARISON

DissipationEnergy

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

forcing & bottom drag

internal wave drag

Barotropic mode:

1

g

@P0

@t+r · (HU0) = 0

@U0

@t+ f ⇥U0 = �rP0 �

DIT

⇢H+ F0

DIT = �⇢rH1X

n=1

Pn

= �rH ⇥ baroclinic pressure

at z=-HInternal waves:@Um

@t+ f ⇥Um = �rPm �rH

1X

n=1

ImnPn

@Pm

@t+r ·

�c2mUm

�= �(c2m)0U0 ·rH �rH ·

1X

n=1

Imnc2nUn

Here, Imn(cm(H), cn(H)) and cm(H) are horizontally varying.

barotropic forcing

M

M

LOCAL MODELLING

•Solve with sponge layers (rather than radiation).

(L� i!I) a = b

linear dynamics unknowns u, v, pbarotropic forcing

LOCAL 3-D MODELLING: INTERNAL TIDES

•Calculate average N(z) on domain from World Ocean Atlas, and then from eigenvalue problem. •Equations are discretized to a 2D sparse system:

cm(H(x))

•Calculate the time-harmonic response for internal wave modes Um(x, t) = Re

⇣Um(x)e�i!t

⌘, m=1,2,3,4, via

�i!Pm +r ·⇣c2mUm

⌘= �(c2m)0U0 ·rH

�i!Um + f ⇥ Um = �rPm

mode-coupling terms neglected for

low-order modelling!

M2: NOVA SCOTIA

Ocean depth (m) Internal wave pressure �N/m2

�

Mode 1 Mode 2

Mode 3 Mode 4

M2 Internal wave

pressure�N/m2

�

Bott

om p

ress

ure

(Pa)

Bott

om p

ress

ure

(Pa) HAWAII

M2(19.4 GW)

K1(1.7 GW)

Bott

om p

ress

ure

(Pa)

M2(4 GW)

AleutianIslands

Bott

om p

ress

ure

(Pa)

Bott

om p

ress

ure

(Pa)

M2(4 GW)

K1(39 GW)

M2 CONVERSION: TPXO.4A vs I.T. MODEL

from JGR, 2001

EGBERT AND RAY: TIDAL ENERGY DISSIPATION 22,485

-20

-40

-60

0 50 100 150 200 250 300 350

Figure 2. Shallow seas and deep ocean areas used for integrated dissipation computations. Areas outlined with solid lines, numbered 1-28, include all of the shallow seas where significant dissipation due to bottom

boundary layer drag would be expected. In particular, essentially all of the dissipation in the prior model (Plate 2a) occurs within these areas. Areas outlined with dashed lines and labeled A-I are deep-water areas which show enhanced dissipation in the T/P estimates of Plate 2.

which simulate the interaction of the barotropic tide with to-

pographic features present in the fine grid but not resolved

by the coarser grid. The pairing of red and blue spots around

this unresolved topography implies that these residuals on

average do no net work around any particular feature. We

demonstrate this more explicitly below by computing area

integrals of dissipation for the prior solution. Because the

real ocean has topography that will not be resolved in the

dynamical equations used to estimate currents, we should

expect similar sorts of noise (and areas of negative dissipa-

tion) in all of our estimates of D.

The remaining three panels of Plate 2 show dissipation

maps based on three tidal solutions representative of the dif-

ferent approaches used to estimate tides from T/P data: for-

mal assimilation methods, empirical corrections to a hydro-

dynamic model, and purely empirical solutions. The first

two of these maps (Plates 2b and 2c) are the TPXO.4a and

GOT99hf estimates presented in E-R. The third approach,

a purely empirical solution with no explicit reliance on any

dynamical assumptions, is represented by the DW95 solu-

tion of Desai and Wahr[ 1995] (Plate 2d). For the GOT99hf

and DW95 solutions, volume transports were computed us-

ing the least squares approach of section 4.2, with the lin-

ear friction parameter of (22) set to a relatively high value

(r = 0.03), and the weight w c of (25) chosen so that

-V. U/iro agrees with the tidal elevation •' to within about

1 cm over most of the open ocean [Ray, 2001 ].

The dissipation map from TPXO.4a has the cleanest ap-

pearance, while the map from the purely empirical DW95

solution is noisiest, with significant areas of negative dissi-

pation throughout the ocean. However, all three maps have

many features in common. The areas of intense dissipation

expected due to bottom boundary layer drag in the shallow

seas (e.g., Patagonian Shelf, Yellow Sea, northwest Aus-

tralian Shelf, European Shelf) are clearly evident in all cases

(compare to the prior solution of Plate 2a). The areas of

enhanced open ocean dissipation discussed in E-R are also

seen in maps for all three T/P-constrained solutions (but not

for the prior solution). For example, dissipation is clearly

enhanced in the Pacific over the Hawaiian Ridge, the Tu- amotu archipelago, and over the back arc island chains ex-

tending from Japan southward to New Zealand. The west-

ern Indian Ocean around the Mascerene Ridge and south of

Madagascar also exhibits enhanced dissipation in all three

of the maps. The Mid-Atlantic Ridge shows up most clearly in the assimilation estimate but is evident also in the other

two estimates. In all three cases there is substantial dissi-

pation throughout much of the North Atlantic. All of these

areas where dissipation is consistently enhanced are charac-

terized by significant bathymetric variation, generally with

Plate 1. Terms in the time averaged energy balance equation (12), derived from the assimilation solution TPXO.4a. In Plates la through ld the work terms are subdivided as in (16)-(19). (a) The work done on the ocean by the large-scale gravitational potential, and (b) the work done by potential terms associated with loading and self-attraction. (c) and (d) Work done by the moving bottom on the ocean, associated with the body tide •'b, and load tide •'t, respectively. (e) The sum of the four work terms plotted in Plates la through ld, and (f) the divergence of the energy flux V ß P. Note that Plates le and I f nearly cancel; the difference is used to estimate the oceanic tidal energy dissipation rate.

Egbert+Ray Model(GW) (GW)

A: Micronesia/Melanesia 100 72C: Mid-Atlantic Ridge 103 47F: Polynesia 31 39H: Hawaii 18 19

Not enough modes for rough

topography; see Zilberman

et al. (2009), JPO

M2: BAROCLINIC CONVERSION

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)

!1000

!100

!10

!1

0

1

10

100

1000

mW

m�2

K1: BAROCLINIC CONVERSION

!1000

!100

!10

!1

0

1

10

100

1000

mW

m�2

39 GW30 GW

55 GW

18 GW

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

•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.

•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).

Calculate surface tide, globally, with

bottom friction

Calculate internal tide

(modal decomposition, 100+ subdomains)

Calculate internal tide drag

and add to barotropic equations

• 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.

CONVERGENCE!

2 4 6 8 10 12 14 16 1810−2

10−1

100

101

iteration number n

max

| h n−

h n−1 |

m2

Barotropic-onlyiterations over

nonlinear friction and self-attraction & loading

(2min each)

Switch on internal tides

Fully coupled iterations

(5hr each, in serial)

A single constituent solution (M2)

A CONVERGING SOLUTION

iteration number

diffe

renc

e in

sol

utio

n

5. SUMMARY AND OUTLOOK

•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.

•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.