Modelling the tides on the North West European Shelf.
Date: 2015-04-13
Institutions: Institute for Environment and Sustainability
Subjects: Atlantic Ocean, earth sciences, geophysics, oceanography, research report, tidal energy

Modelling the tides on the North 
West European Shelf 

Clare Coughlan 
Adolf Stips 
2 0 1 5  

Report EUR 27114 EN 

 
 
European Commission 
Joint Research Centre 
Institute for Environment and Sustainability 

Contact information 
Adolf Stips 
Address: Joint Research Centre, Via Enrico Fermi 2749, TP 270, 21027 Ispra (VA), Italy 
E-mail: adolf.stips@jrc.ec.europa.eu 
Tel.: +39 0332 78 9876 
Fax: +39 0332 78 5601 

JRC Science Hub  
https://ec.europa.eu/jrc 

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expressed does not imply a policy position of the European Commission. Neither the European Commission nor any person 
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All images © European Union 2015, except: cover foto (©ISTOCK) 

JRC93441 

EUR 27114 EN 

ISBN 978-92-79-45680-0 (PDF) 

ISSN 1831-9424 (online) 

doi:10.2788/17992 

Luxembourg: Publications Office of the European Union, 2015 

© European Union, 2015 

Reproduction is authorised provided the source is acknowledged. 

Abstract 

Tides are a key process in the dynamics of the North West European Shelf. A GETM model has been developed for the 
region  and  this  report  describes  the  model  performance.  Measured  harmonic  constituents  are  compared  with  model 
outputs and these results are put into context with other shelf sea models of the region. Most of the differences between 
the model and observations are within the errors that are expected for a shelf sea model, and the overall statistics are 
skewed by poor performance in a few places. The major constituents are not represented particularly well in the Irish Sea, 
Celtic Sea and English Channel regions, but overall this model performs reasonably well, and better than many other shelf 
sea models of the region. 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
              Contents 
Introduction ...................................................................................................................................................... 3 
North West European Shelf GETM Model ............................................................................................. 4 
Representation of tides in the model ................................................................................................. 5 
Experimental setup ........................................................................................................................................ 6 
Results ................................................................................................................................................................ 8 
Conclusions .................................................................................................................................................... 22 
References ...................................................................................................................................................... 23 

Introduction 
The  North  West  European  Shelf  is  a  broad  temperate  shelf  forming  the  eastern  margin  of 
the  Northern  North  Atlantic.  It  includes  several  shelf  sea  regions  that  are  adjacent  to  the 
most  populous  and  industrialized  countries  of  Europe.  The  dynamics  of  the  region  are 
controlled  by  the  seasonal  heating  cycle,  atmospheric  fluxes,  tides,  river  inputs  and 
exchanges  with  the  open  ocean.  The  region  can  largely  be  described  as  a  seasonally 
stratified,  downwelling  shelf  sea  system  (Holt  et  al.,  2009)  with  a  net  inflow  of  surface 
waters and a net outflow on the sea floor across the shelf break into the deep Atlantic with a 
generally  anti-clockwise circulation  of  the North Sea  (Wakelin  et  al.,  2008).  The  large  scale 
ocean-shelf exchange is controlled by seasonal upwelling in the south of the region (Gomez-
Gesteira  et  al.,  2011),  and  the  poleward  slope  current  and  Ekman  transport  in  the  North 
(Holt  et  al.,  2009;  Huthnance  et  al.,  2009).  Tidal  mixing  fronts  separate  the  seasonally 
stratified  from  the  well  mixed/sporadically  stratified  shallower  regions  either  nearer  the 
coast  or  on  banks  and  shoals.  Although  tides  play  a  significant  role  in  the  circulation, 
horizontal  density  gradients  play  an  important  role,  particularly  during  the  summer  and 
autumn months (Holt & Proctor, 2008). 

The  region  is  further  characterized  by  two  strong  northward  currents  (the  slope  currents 
along  the  shelf  break  and  the  current  along  the  Norwegian  trench),  that  enclose  the 
Northern part of the shelf. Local conditions in the area can be highly heterogeneous due to 
strong tidal dynamics, riverine inflow and the interface with the brackish waters of the Baltic 
Sea.  Much  of  the  open  shelf  is  seasonally  stratified;  tidal  mixing  fronts  separate  stratified 
regions from well-mixed/sporadically stratified shallower regions. Tidal mixing fronts tend to 
separate nutrient-depleted from nutrient-rich waters, and cross frontal exchange processes 
can result in enhanced biological production (Pingree & Griffiths, 1982; Richardson & Visser, 
2000),  making  these  regions  important  for  shelf  sea  coupled  hydrodynamic  ecosystem 
modelling.  Since  the  extent  of  stratified  regions  is  dependent  on  the  local  relationship 
between  depth  and  tidal  mixing  (Simpson  &  Hunter,  1974),  it  is  important  that  the  model 

 
 
 
accurately  reproduces  the  main  tidal  constituents.  The  vertical  temperature  structure, 
through  its  control  over  mixing,  is  crucial  to  the  prediction  of  the  vertical  distribution  of 
nutrients  and  contaminants,  with  consequent  implications  for  biological  and  water  quality 
modelling.  

The  North  West  European  Shelf  has  some  of  the  largest  tides  on  the  globe:  for  example, 
Avonmouth (Bristol Channel) has a spring tidal range of over 14 m (Pelling et al., 2013). The 
large  tides  make  this  one  of  the  most  energetic  areas,  with  total  dissipation  rates 
approaching  200  GW  (Egbert  &  Ray,  2001),  which  is  about  5-6%  of  the  total  present-day 
global  tidal  dissipation.  Tides  across  the  region  are  dominated  by  the  semi-diurnal 
constituents  (in  particular  M2  and  S2,  with  M2  ca.  30-50%  of  S2  according  to  Uehara  et  al., 
2006).  The predominantly  semi-diurnal  tides  (Pingree  &  Griffiths, 1982) are  a  co-oscillating 
response of the shelf seas to the tides generated in the Atlantic Ocean. Tidal energy from the 
Atlantic  is  transmitted  onto  the  European  shelf  into  the  Celtic  Sea  between  Brittany  and 
southern Ireland via the Atlantic semi-diurnal Kelvin wave, which travels south to north. The 
wave  propagates  into the  English  Channel  and  energy passes  into  the  southern North  Sea, 
the Irish Sea and into the Bristol Channel (Pugh, 1996). The north of Scotland diffracts part of 
the semi-diurnal wave, and it turns east and to the south into the North Sea. The diurnal tide 
behaves as a standing wave in the Celtic Sea, the Bristol Channel and English Channel, but 
without  any  tendency  to  resonance  (Pugh,  1996).  The  tides  in  the  North  and  Irish  seas 
essentially behave like a Kelvin wave that enters from the North Atlantic, propagates along 
the UK coast, and is reflected at the end of each basin. The reflected wave then travels back 
on the opposite side of the basin (i.e. along the coast of continental Europe in the North Sea 
or the Irish Coast in the Irish Sea). 

North West European Shelf GETM Model 
The  3  minute  x  3  minute  North  West  European  Shelf  (NWES)  hydrodynamic  model  is  an 
implementation of the General Estuarine Transport Model, GETM (http://getm.eu, Burchard 
and  Bolding,  2002;  Stips  et  al.,  2004).  GETM  is  a  three-dimensional  free-surface  primitive 
equation  Arakawa  C-grid  (Arakawa  &  Lamb,  1977)  model  that  uses  the  Boussinesq  and 
boundary layer approximations. The Arakawa C-grid is known to be particularly prone to grid 
scale noise due to spatial averaging of Coriolis terms, but provided the deformation radius is 
well-resolved (~30 km), C-grid models yield the most accurate numerical solutions (Adcroft 
et al, 1999). The free-surface, density and active/passive tracers are located at the centre of 
the  cell,  whereas  the  horizontal  velocities  (u  and  v)  are  located  at  the  west/east  and 
south/north edges of the cell, respectively. 

The  model  domain  extends  from  the  deep  ocean  to  the  coast  from  46.4˚N  to  63˚N  and 
17.5˚W to 13˚E and is divided into a 1/20˚ grid with (i, j) = (1, 1) at the south-west corner and 
dimensions (308, 347). The domain is large and the resolution is limited by the desire to be 
4 

 
 
able  to  produce  results  in  a  reasonable  time  frame.  Horizontal  spherical  coordinates  and 
vertical,  terrain-following  σ-coordinates  are  combined  to  give  a  grid  spacing  of  ~6  km 
resolution with 25 layers in the vertical. The σ-coordinates are equidistant for water depths 
shallower than  150  m,  but  in deeper  water the  levels  are  concentrated at  the  surface  and 
the  bed  using  a  generalised  version  of  the  mixed–layer  transformation  proposed  by 
(Burchard & Petersen, 1997), in order to better resolve the surface mixed layer and bottom 
boundary layer.  

The internal Rossby radius in this domain might be expected to range from ~3 km in the river 
plumes to 10-20 km in the Norwegian Trench to ~30 km in the northeast Atlantic. This model 
is not expected to fully resolve the details of the on-shelf baroclinic features (such as frontal 
instabilities  and  river  plumes),  but  it  is  hoped  that  their  overall  characteristics  (such  as 
frontal locations) are captured. Ideally the model would be of sufficient resolution to resolve 
both the internal and external radii, i.e. a resolution of the order <2 km, but at present the 
computational  cost  of  such  a  system  makes  this  impractical  for  coupled  hydrodynamic-
ecosystem modelling over an area of this extent.  

A  crucial  issue  with  model  development  is  balancing  accuracy  (how  well  the  model 
reproduces  in  situ  data)  with  respect  to  its  ability  to  reproduce  temporal  trends  (for 
example, how closely it reproduces the observed seasonal cycle. The North West European 
Shelf is an exceptionally data rich region and has been extensively modelled. Verification of 
the  3nm  NWES  model  is  achieved  by  comparison  with  a  variety  of  observations,  gridded 
products  and  other  model  simulations  to  evaluate  its  ability  to  reproduce  instantaneous 
point  values  (and  trends)  and  spatial  characteristics  (and  trends).  The  model  domain 
encompasses both open ocean and shelf sea regions, presenting a sizable challenge for any 
model. 

Representation of tides in the model 
Current and sea surface elevation within the model domain are driven by Flather boundary 
conditions  (Flather,  1976;  Carter  &  Merrifield,  2007).  The  Flather  condition  is  a  radiation 
boundary  condition  that  combines  the  Sommerfield  equation  (with  surface  gravity  wave 
phase  speed)  with  a  one-dimensional  version  of  the  continuity  equation  applied  in  the 
outwardly normal direction at an open boundary: 

represents  the  external  data,  h  is  the  local  water  depth  and , 

and  are  the 
prognostic variables prescribed along the boundary for the incoming wave. The differences 
between the external data and the model predictions are allowed to propagate out of the 
domain at the speed of the external gravity waves. Volume is conserved in the domain and 
variations  due  to  physical  forcing,  such  as  tides,  can  be  introduced  through  the  external 
5 

extextnnhgvvextnvuv 
 
data. Since the open boundary condition involves only the depth-mean current, any current 
structure within the model is generated by the physics, i.e. primarily frictional effects (bed 
friction and internal friction). For each of the model open boundary points, hourly elevation 
and  current  data  were derived from the  Oregon  State  University inverse  modelled  netCDF 
gridded  data  set  (Egbert  &  Erofeeva,  2002;  http://volkov.oce.orst.edu/tides/).  Hourly  data 
were required, as with a greater timestep the data did not capture the full signal along the 
boundary, leading to an underestimation of the maximum elevations and velocities.   

Experimental setup 
The two main controls on the tides are the bathymetry and the bottom friction. Isolated cells 
were  found  to  affect  the  tidal  propagation  through  the  model,  so  a  large  amount  of  work 
was put into adjusting the coastline to avoid the creation of ‘lake’ regions during wetting and 
drying. A range of values for the coefficient of bottom friction (z0) was tested in the model to 
calibrate the tidal harmonics. The ability of the model to reproduce the tidal regime across 
the  North  West  European  Shelf  is  assessed  by  comparing  the  modelled  harmonic 
constituents  against  those  derived  from  observations  at  tide  gauges  and  current  measure, 
and FES2012 atlases (Finite Element Solutions), a tide model that includes data assimilation 
(Carrere 
2012;  http://www.aviso.oceanobs.com/en/data/products/auxiliary-
products/global-tide-fes.html).  The  harmonic  constants  from  the  tide  gauge  and  current 
meter observations were extracted from the literature.  

al., 

et 

The FES2012 solution is obtained through the use of a finite-element hydrodynamic model 
that  assimilates  long-term  altimetry  data  (Topex/Poseidon,  Jason-1,  Jason-2,  ERS-1,  ERS-2 
and ENVISAT) and in situ harmonic data; although only 3 tide gauges have been assimilated 
so  far  for  the  European  Shelf  region.  32  tidal  constituents  are  distributed  on  1/16˚  grids 
(amplitude and phase). It is considered a fairly reliable reference in the deep ocean (Maraldi 
et al., 2013) and good results are obtained in shelf regions (Carrere et al., 2012). 

Initial model runs were made using a constant bed roughness. These preliminary tests were 
run over 5 different bathymetries: 

•  B1: bathymetry base on NOOS 

•  B2: B1 depths increased by 5% throughout the domain 

•  B3: bathymetry base on GEBCO_08 

•  B4: B1 improved in coastal regions (drying ‘lakes’ removed) 

•  B5: B4 depths increased by 8% or a maximum of 5 m (where depths were greater 

than 62.6 m) 

6 

 
 
Davies & Aldridge (1993) examined the influence of small changes in water depth upon tidal 
elevations  and  currents  in  the  Irish  Sea.  They  found  that  an  increase  in  water  depth  by  a 
factor of 1.08 significantly improved the accuracy of the elevations and current magnitudes 
computed with a three-dimensional model. The justification for the increase in water depths 
is that depths from navigation charts tend to be biased towards minimum depths to avoid 
vessels going aground.  

Observed values of friction coefficient range from 4.30 x 10-3 to 5.29 ± 1.7 x 10-3 (reported in 
Davies, 1986). Bed roughness values used in other European Shelf models range from 0.001 
to  0.035  (see  Table  1).  These  values  are  larger  than  the  mean  values  for  different  bottom 
types reported in, for example, Soulsby, 1983. The larger values take into account the form 
drag over sand waves that cannot be resolved in larger grid models. No other models in this 
region have implemented a variable bed roughness. Bed roughness was varied from 0.001 to 
0.008,  corresponding  to  the  sorts  of  values  used  in  other  shelf  models  and  from 
observations.  Initially,  a  spatially  constant  value  of  bed  roughness  was  applied  across  the 
domain,  but  later  tests  were  conducted  with  a  spatially  varying  bed  roughness  to  try  to 
improve  regions  with  a  poor  match  between  the  model  and  observations.  The  overall 
statistics  comparing  the  modelled  harmonic  elevation  to  those  derived  from  observations 
are summarised by the Taylor diagrams (Taylor, 2001) shown in Figure 1 to Figure 5. While 
these  runs  were  being  completed,  the  GETM  version  was  updated  from  2.2  to  2.4.  The 
naming convention for the runs is [bathymetry]_[z0 value]_[getm version]. 

Table 1: Values of z0 used in other models 

z0 

Reference 

Region 

0.0025 

Xing et al. (2011); Davies et al. (1997) 

European Shelf 

0.003, 0.0025, 0.00125 

Davies et al. (2011)  

Celtic and Irish Sea 

0.0035 

Maraldi et al. (2013) 

Iberia-Biscay-Ireland 

0.005 

Davies & Kwong (2000); Davies (1986) 

European Shelf 

0.015, 0.025, 0.035 

Davies & Jones (1990) 

European Shelf 

7 

 
 
 
 
 
 
Results 
Taylor  diagrams  provide  a  summary  of  overall  model  performance,  compared  to  the 
observations. The grey contours show the RMS errors and the dashed arc gives the observed 
standard deviation. The star shows the observed value. The distance from the origin to each 
model point is the standard deviation of the modelled amplitudes or phases; points closer to 
the  origin  than  the dashed  line underestimate the  variance  of the  data,  and  those  outside 
the  dashed  line  overestimate  this  variance.  In  general  the  model  captures  the  pattern  of 
variation, with similar standard deviations between observed and modelled amplitudes and 
phases, except for the smaller constituents, where the magnitude of the spatial variability is 
either  under-  or  over-  represented  in  the  model,  depending  on  the  model  run.  The  major 
constituents  (M2  and  S2)  have  a  high  correlation  with  the  phase  and  amplitude  of  the 
observed harmonic constituents, but the RMS errors are still fairly large. The bathymetries 
with  the depth  increase  (B2,  B5)  gave  the  worst  comparisons  with  M2  amplitudes. Varying 
the  bathymetry  and/or  the  bottom  roughness  does  not  improve  the  match  between  the 
modelled  and  observed  S2  amplitudes,  only  increases  or  decreases  the  spatial  variability 
between the points.  

Figure 1: Taylor diagram showing the overall model performance for varying bathymetry, bed roughness and 
GETM version for the M2 tidal amplitude (left) and phase (right) 

Correlations  are  high,  apart  from  the  phase  of  the  M4  tide.  The  K1  constituent  is  not 
particularly well-modelled. However, this is a small constituent (of the order of less than a 
few centimetres per second), so it is difficult to make meaningful comparisons. There is large 
spatial variability in the harmonic constituents, so the Taylor plots do not show any trends in 
model performance for tuning the bathymetry and bottom friction. 

8 

 
 
 
 
Figure 2: as for Figure 1, but the S2 tide 

Figure 3: As for Figure 1, but the K1 tide 

Figure 4: as for Figure 1, but the O1 tide 

9 

 
 
 
 
Figure 5: as for Figure 1, but the M4 tide 

Summary statistics were compared with other model runs found in the literature to assess 
the order of the errors. Table 2 gives a short summary of the models;  

Table 3 shows the model statistics for the major diurnal constituents and Table 4 the major 
semi-diurnal  statistics.  The  M4  statistics  are  also  shown  in  Table  4.  This  constituent  is  not 
forced at the boundary, but is calculated in the model due to bathymetric effects. Note that 
due to the different model resolutions and domains, not all of the comparisons below will 
have  been  with  the  same  number  of  observations  in  exactly  the  same  locations,  but  this 
comparison gives an idea of the order of the errors that are found in other European Shelf 
models. 

10 

 
 
 
 
 
 
 
Table 2: Tidally validated models found in the literature 

Abbrv. 

Reference 

Short description 

PD96 

Proctor & Davies (1996) 

B grid, σ-coords, POLCOMS 

K97 

H01 

H05 

Kwong et al. (1997) 

C grid, 1/6˚x 1/9˚ 

Holt et al. (2001) 

B grid s-coords, POLCOMS, 1/6˚x 1/9˚ 

Holt et al. (2005) 

B grid, s-coords, POLCOMS, 1/10˚x 1/7˚ 

FOAM12 

O’dea et al. (2012) 

s-σ cords, NEMO, 1/15˚x 1/9˚ 

IBI13 

Maraldi et al. (2013) 

C grid, NEMO, 1/36˚x 1/36˚ 

C14 

This model 

C grid, GETM, 1/20˚x 1/20˚ 

Table 3 and Table 4 give the mean, RMS error and error in the complex plane (Hs). The mean 
error  is  simply the  mean  of  the differences  between  the  modelled and  observed  values. A 
is  model 
positive  value 
underprediction. The root mean square error (RMSE) gives an idea of the size of the model 
errors in physical terms: 

indicates  model  overestimation  and  a  negative  value 

 is  the  number  of  data  points  and 

 and 

 are  observed  and  modelled  values, 

respectively. Hs combines both amplitude and phase error into a single error measurement 
(Kwong et al., 1997). 

and 

are the observed and computed amplitude and phase at each gauge, and 

is the number of locations where observational data are available.  

11 

nOMRMSEniii12niOiMniiisSCnH121221)cos()cos(00ccighghC)sin()sin(00ccighghS00,ghccgh,n 
 
 
 
 
 
 
 
 
Table 3: A comparison of modelled diurnal tidal constituents with tide gauge measurements. Statistics are from 
model  minus  observed  values  of  amplitude  (h  cm)  and  phase  (g˚).  Hs  is  the  error  in  the  complex  plane  (cm) 
defined in Kwong et al. (1997) 

Statistic 

Model 

Q1 

O1 

K1 

h 

g 

h 

g 

h 

g 

PD96 

-0.40 

12.7 

-2.00 

3.7 

1.60 

-16.4 

K97 

-0.90 

3.3 

-1.50 

-6.4 

-0.70 

-11.5 

H01 

-0.41 

10.2 

-1.96 

-5.6 

2.95 

-19.3 

Mean 

H05 

0.00 

FOAM12 

- 

- 

- 

2.00 

- 

-1.30 

- 

-1.30 

-2.2 

-0.20 

-8.4 

This 
model 

-0.32 

-11.4 

-0.45 

-10.2 

0.66 

5.47 

PD96 

6.97 

42.25 

2.74 

20.67 

2.92 

34.49 

K97 

6.98 

39.60 

2.22 

24.20 

1.88 

26.30 

H01 

6.97 

42.12 

2.68 

24.10 

4.23 

36.50 

RMS 

H05 

6.8 

FOAM12 

- 

- 

- 

2.60 

- 

2.40 

- 

1.90 

15.70 

1.80 

17.70 

0.60 

27.74 

1.10 

29.31 

1.80 

21.25 

3 

2.7 

3.08 

1.30 

1.89 

4.3 

2.8 

5.53 

1.8 

2.33 

This 
model 

PD96 

K97 

H01 

Hs 

1.90 

1.60 

1.92 

FOAM12 

- 

This 
model 

0.89 

12 

 
 
Table 4: A comparison of modelled semi-diurnal tidal constituents with tide gauge measurements. Statistics are 
from  model  minus  observed  values  of  amplitude  (h  cm)  and  phase  (g˚).  Hs  is  the  error  in  the  complex  plane 
(cm) defined in Kwong et al. (1997) 

Statistic 

Model 

N2 

M2 

S2 

M4 

h 

g 

h 

g 

h 

g 

h 

g 

PD96 

1.5 

-3.6 

-3.3 

1.3 

0.9 

-3.8 

-0.5 

26.3 

K97 

-0.1 

-11.1 

-2.4 

-2.4 

-1.5 

-9.5 

-1.5 

0.6 

H01 

1.44 

-5.48 

4.99 

Mean 

H05 

-1.6 

- 

-1.9 

1 

- 

3.83 

-5.31 

1.33 

26.3 

-1.7 

- 

-1.8 

FOAM12 

0.4 

2.7 

-4.7 

-0.2 

-0.4 

0.5 

- 

- 

- 

This 
model 

0.08 

-1.91 

-0.1 

-3.2 

0.28 

-3.55 

-0.85 

8.28 

PD96 

4.5 

26.04 

14.25 

14.99 

6.54 

22.3 

4.7 

82.17 

K97 

3.9 

28.9 

12.7 

17.3 

7.32 

25.1 

5.48 

80.9 

H01 

4.52 

26.68 

14.9 

14.76 

7.64 

22.55 

5.38 

82.55 

RMS 

H05 

4.6 

- 

16.3 

- 

7.3 

- 

5.0 

FOAM12 

2.9 

21.6 

10.3 

14.7 

3.7 

12.8 

- 

- 

- 

This 
model 

PD96 

K97 

H01 

IBI13 

This 
model 

Hs 

2.8 

16.14 

18.0 

13.4 

4.6 

17.90 

3.6 

48.52 

6.9 

5.9 

6.79 

- 

3.98 

21.1 

16.6 

21.61 

21.6 

19.85 

11.2 

10.2 

12.04 

8.0 

6.99 

5.9 

5.4 

6.84 

7.1 

4.15 

13 

 
 
 
 
As seen in Table 3 and Table 4, the model errors are typical for models of this region. The 
RMS  error  for  the  M2  elevation  is  higher  than  for  other  models,  but  the  phase  is  better 
modelled,  which  give  a  better  overall  Hs  value.  The  M4  tide,  which  is  not  forced  on  the 
boundary,  but  produced  through  interaction  of  the  M2  with  the  bathymetry,  is  modelled 
much better in this model than in other shelf models. In fact, all constituents are modelled 
better than other shelf models, apart from K1, which is a minor constituent with very small 
amplitudes.  Errors  of  the  order  shown  here  in  the  diurnal  tides  are  to  be  expected,  and 
relate to the accuracy with which these constituents can be measured. 

The Taylor plots only give an impression of how the model is performing overall, and spatial 
plots  are  used  to  determine  where  the  model  is  performing  well  and  where  it  can  be 
improved.  Amphidromes  represent  the  nodes  of  standing  waves  in  a  rotating  system,  and 
are  formed  by  the  interaction  of  incident  and  reflected  Kelvin  waves.  Their  position  is 
affected by water depth, frictional effects and topography, and the model does a good job of 
predicting their locations. The semi-diurnal constituents give similar co-tidal patterns (Figure 
6),  although  the  M2  tide  has  by  far  the  largest  amplitudes.  There  is  a  degenerate 
amphidromic point at the southern tip of Norway with an amphidromic point off the west 
coast  of  Denmark  and  another  in  the  Southern  Bight  of  the  North  Sea.  The  amphidromic 
points  between  Scotland  and  Northern  Ireland  and  off  south-east  Ireland  are  degenerate. 
The pattern in the semi-diurnal co-tidal charts (Figure 6) shows larger spatial variability than 
that  found  for  the  diurnal  tides  (Figure  7)  as  a  result  of  the  shorter  wavelength  of  these 
components.  The  diurnal  co-tidal  charts  (Figure  7)  are  characterised  by  an  amphidromic 
point  off  the  south-west  corner  of  Norway  with  tidal  amplitudes  increasing  to  the  south-
west of this point.  

Figure 6: Co-tidal plots for the major harmonic constituents, M2 (left) and S2 (right). The labelled black lines give 
the tidal elevation and the coloured lines the phase 

14 

 
 
 
Figure 7: Co-tidal chart for the M2 tidal constituent based solely on observational data from tide gauges and 
bottom pressure sensors (from Howarth, 1990). The dashed lines with values are lines of constant amplitude 
(m); solid lines indicate lines of constant phase 

Figure 6 demonstrates that the model reproduces the expected pattern of propagating tidal 
waves around amphidromic points, with similarities in the spatial variation to that found in 
other  shelf-wide  models  (O’dea  et  al.,  2012;  Holt  et  al.,  2005;  S.  C.  M.  Kwong  et  al.,  1997) 
and observations (Howarth, 1990, see Figure 7). In particular, the degenerate amphidrome 
south of Norway that has proven to be difficult to obtain in other modelling studies  (O’dea 
et  al.,  2012)  is  well-modelled.  Figure  8  shows  the  location  of  this  point  in  the  FOAM  and 
POLCOMS  models  (O’dea  et  al.,  2012).  The  amphidromic  point  off  Norway  is  displaced 
offshore (westwards) for decreasing z0 (not shown here), but there is no significant change 
in the location of the other amphidromic points for changing z0. The increase in amplitudes 
from ocean to shelf and within the English Channel and Southern Bight of the North Sea is 
clearly evident. 

15 

 
 
 
Figure 8: M2 co-tidal chart for FOAM12 (left) and POLCOMS (right). Taken from O'Dea et al., 2012 

Figure 9: As for Figure 6, but for the major diurnal constituents, Q1 (left) and O1 (right) 

The errors are not distributed evenly across the domain. In Figure 10 to Figure 14 the filled 
contours  show  the  differences  between  modelled  output  and  the  FES2012  atlas  and  the 
coloured circles give differences between modelled output and tide gauges (the scale is the 
same  and  these  are  absolute  values  of  model  –  observation,  where  blue  indicates  model 
underprediction and red is model overprediction). The tides tend to be well modelled in the 
open sea regions (for example, Figure 13), however, there is an overprediction in the Celtic 
Sea,  western  English  Channel,  the  North  Channel  approach  to  the  Irish  Sea  and  the  Baltic, 
and an underprediction in the Irish Sea.  

16 

 
 
 
 
 
Figure 10: Distribution of errors from a comparison of M2 elevation amplitude (left) and phase (right) for the B3 
GEBCO_08 bathymetry (6 months harmonic analysis); z0 = 0.001 

Figure 11: Distribution of errors from a comparison of M2 elevation amplitude (left) and phase (right) for the B1 
NOOS bathymetry (6 months harmonic analysis); z0 = 0.001 

17 

 
 
 
 
 
 
Figure 12: Distribution of errors from a comparison of M2 elevation amplitude (left) and phase (right) for the B1 
NOOS bathymetry (6 months harmonic analysis), z0 = 0.003 

Figure 13: Distribution of errors from a comparison of M2 elevation amplitude (left) and phase (right) for the 
B1 NOOS bathymetry (6 months harmonic analysis), z0 = 0.005 

Figure 14: Distribution of errors from a comparison of M2 elevation amplitude (left) and phase (right) for the 
B1 NOOS bathymetry (6 months harmonic analysis), z0 = 0.008 

A large part of the of the mean amplitude error for M2 is due to an underestimation of the 
M2 amplitude  in  the  Irish  sea,  as  found  in  other  shelf  models  (for  example,  O’Dea  et  al., 
2012). The Irish and Celtic Seas are particularly problematic with underestimation of the M2 
tide in the Irish Sea and overestimation in the approaching channels (North Channel and St 
George’s  Channel).  This  may  be  due  to  a  number  of  factors,  including  errors  in  boundary 
input  and  bathymetry,  dissipation  being  too  strong  in  regions  of  underestimation,  or  too 
weak in regions of overestimation. The model grid may be too coarse to resolve regions such 
as the North Channel and Bristol Channel.  

18 

 
 
 
 
 
The GEBCO_08 bathymetry produces reasonable statistics overall (the best overall statistics 
for all the runs with spatially constant z0, and Figure 10 shows that the model matches the 
observations  in  many  regions.  The  Skagerrak  region  is  much  better  modelled  than  for  the 
NOOS bathymetry (for example, Figure 11). However, most of the English Channel and Celtic 
Sea (where there are no observations) is poorly modelled and there are much larger phase 
differences than seen when using the NOOS bathymetry. 

With the NOOS bathymetry, the model tends to overpredict tidal elevations in the western 
English Channel and Celtic Sea, the northern approaches to the Irish Sea and the Skagerrak 
region.  Elevations  in  the  Irish  Sea  are  underpredicted.  The  phase  is  reasonably  well-
modelled,  but  the  wave  tends  to  be  travelling  too  slowly  along  the  German  and  Dutch 
coastline.  A  bottom  friction  of  0.001  is  too  low,  and  the  elevations  are  overpredicted  for 
most of the coastal parts of the domain. Varying z0 does not have much effect on the phase, 
which is fairly well represented across the domain. None of the values tested improve the 
issue  of  underprediction  in  the  Irish  Sea,  but  a  constant  value  of  0.005  gives  the  best 
representation across the domain. These are the values shown in  Table 3 and Table 4, and 
although the RMSE for M2 elevations is slightly larger than other shelf models, all the other 
constituents are modelled better. A full set of statistics for the model run with z0 = 0.005 is 
shown in Table 5 and Table 6. 

Table 5: Modelled statistics for semi-diurnal tidal elevations for the NOOS bathymetry with z0 = 0.005 

Statistic 

M2 

S2 

M4 

h 

g 

h 

g 

h 

g 

RMSE 

18.00 

13.35 

4.60 

17.90 

3.60 

48.52 

Reliability 
index 

1.29 

1.75 

1.19 

1.18 

2.09 

2.95 

NSME 

0.95 

0.98 

0.95 

0.93 

0.74 

0.77 

pbias 

-0.04 

-1.65 

0.87 

-1.79 

-10.97 

3.96 

Cost 
function 

Corr 
coeff 

-0.00 

-0.04 

0.01 

-0.05 

-0.12 

0.08 

0.98 

0.95 

0.98 

0.97 

0.88 

0.58 

19 

 
 
 
 
 
 
 
Table 6: Modelled statistics for diurnal tidal elevations for the NOOS bathymetry with z0 = 0.005 

Statistic 

K1 

O1 

h 

g 

h 

g 

RMSE 

1.80 

21.25 

1.10 

29.31 

Reliability 
index 

1.41 

30.16 

1.35 

4.56 

NSME 

0.33 

0.95 

0.79 

0.95 

pbias 

10.13 

-3.24 

-7.20 

5.18 

Cost 
function 

Corr 
coeff 

0.30 

-0.05 

-0.19 

0.08 

0.74 

0.69 

0.91 

0.43 

20 

 
 
 
 
Table 7: RMS error of the complex amplitude difference between observed and modelled tidal components for 
the  sea  surface  elevation.  Units  are  in  cm.  The  Irish  Sea  is  delimited  by  51˚N–56˚N,  9˚W-3˚W;  the  English 
Channel is delimited by 48.5˚N-51.5˚N; the North Sea region is delimited by 51.5˚N-60˚N, 4.5˚W-7.5˚E; and the 
Baltic  region  is  delimited  by  56˚N-59˚N,  7.5˚E-13˚E.  Maraldi  et  al.  (2013)  examined  these  same  regions, 
although their model covers a much wider region. 

Region  Model run 

M2 

S2 

K1 

O1 

M4 

Whole 
domain 

Irish 
Sea 

English 
Channel 

North 
Sea 

Baltic 
Sea 

Maraldi et 
al., (2013) 

21.6 

8.0 

1.8 

1.3 

7.1 

z0 = 0.005 

19.85 

6.99 

2.33 

1.89 

4.15 

varyz0 

19.73 

7.00 

2.42 

1.91 

4.17 

varyz0b 

20.08 

7.01 

2.24 

1.94 

4.14 

varyz0c 

20.13 

7.01 

2.43 

1.92 

4.17 

Maraldi et 
al., (2013) 

25.9 

10.1 

1.7 

1.3 

11.1 

z0 = 0.005 

32.30 

16.10 

2.23 

2.60 

4.32 

varyz0 

32.59 

15.70 

2.28 

2.64 

4.42 

varyz0b 

33.87 

15.83 

2.24 

2.69 

4.34 

varyz0c 

33.90 

15.80 

2.28 

2.66 

4.43 

Maraldi et 
al., (2013) 

23.6 

8.1 

1.5 

1.9 

7.5 

z0 = 0.005 

39.67 

15.13 

1.71 

1.86 

9.70 

varyz0 

36.82 

15.77 

1.78 

1.89 

10.15 

varyz0b 

36.81 

14.89 

1.91 

1.85 

10.20 

varyz0c 

37.29 

15.66 

1.76 

1.88 

10.15 

Maraldi et 
al., (2013) 

14.7 

4.7 

2.7 

1.0 

6.2 

z0 = 0.005 

20.17 

4.37 

2.84 

1.98 

4.26 

varyz0 

20.90 

4.42 

2.94 

1.98 

4.20 

varyz0b 

21.52 

4.34 

2.65 

1.95 

4.12 

varyz0c 

21.26 

4.41 

2.94 

1.98 

4.20 

Maraldi et 
al., (2013) 

29.8 

9.0 

2.4 

2.1 

0.9 

z0 = 0.005 

13.5 

2.95 

0.30 

1.13 

2.71 

varyz0 

13.45 

3.05 

1.45 

1.27 

2.57 

varyz0b 

10.35 

2.56 

1.17 

1.48 

2.35 

varyz0c 

13.15 

3.02 

1.44 

1.26 

2.54 

21 

 
 
Tests were made with varying the bed roughness to try to improve the model results in the 
Celtic Sea and the English Channel (not shown here). This did not lead to significant changes 
in the overall patterns and tidal elevations in the Irish Sea are still underpredicted. A spatially 
constant z0 = 0.005 gives the best overall match with observations.  

Table  7  gives  a  regional  comparison  of  the  RMS  of  the  complex  difference  (Kwong  et  al., 
1997)  for  each  tidal  constituent.  To  appreciate  how  significant  these  errors  are,  they  are 
compared  with  overlapping  regions  assessed  by  Maraldi  et  al.  (2013).  Note  however,  that 
the measurement points differ between this study and that of Maraldi et al. (2013). Overall 
this  model  performs  reasonably  well,  but  the  major  constituents  are  not  represented 
particularly well in the Irish Sea, Celtic Sea and English Channel regions. No solution has yet 
been found to improve the model performance in these regions.  

Most of the differences between the model and observations are within the errors that are 
expected for a shelf sea model, and the overall statistics are skewed by poor performance in 
a  few  places.  This  model  performs  much  better  than  the  FES2012  model  for  the  M4 
constituent. This is to be expected due to the better representation of coastal regions in our 
finer  resolution  model.  There  is  still  an  underprediction  of  the  M4  tide  by  this  model, 
suggesting that  the  advective terms  in  near-shore  regions  are  still  not well-represented.  In 
particular, the phase of the M4 tide is not well-matched with the observations. 

Conclusions 
A  three  dimensional  hydrodynamic  GETM  model  has  been  set  up  for  the  North  West 
European Shelf. This is a tidally important region, where tidal motions provide the dominant 
source of energy and therefore tides play a key role in the development of the water column 
properties.  Model  validation  is  essential  to  illustrate  the  potential  use  of  this  model  to 
capture the wide variety of processes and scales across the shelf region, and the 3nm NWES 
GETM model has been shown to capture the complicated tidal dynamics of this region. In a 
few regions (Irish Sea, Celtic Sea and English Channel), the major tidal constituents are not 
well-represented, but for most of the shelf region the model performs well within the errors 
that are expected for a shelf sea model. The analysis of the tidal elevations and current fields 
show  that  bathymetry  is  the  main  control  on  tidal  wave  propagation;  small  errors  in  the 
bathymetry (depth and slope) can generate large displacements in the circulation patterns. 
Further work should focus on improving the local bathymetry in the Irish Sea, Celtic Sea and 
English Channel regions. Overall, this model has been shown to have a good representation 
of the tides across the shelf and is therefore provides a good tool for further studies in the 
region. 

22 

 
 
 
 
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25 

 
 
 
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EUR 27114 EN – Joint Research Centre – Institute for Environment and Sustainability 

Title: Modelling the tides on the North West European Shelf 

Authors: Clare Coughlan, Adolf Stips 

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2015 – 25 pp. – 21.0 x 29.7 cm 

EUR – Scientific and Technical Research series – ISSN 1831-9424 (online) 

ISBN 978-92-79-45680-0 (PDF) 

doi: 10.2788/17992 

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