On the performance of a generic length scale turbulence model within an adaptive finite element ocean model

Ocean Modelling 56 (2012) 1–15

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Ocean Modelling

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On the performance of a generic length scale turbulence modelwithin an adaptive finite element ocean model

Jon Hill a,⇑, M.D. Piggott a,b, David A. Ham a,b, E.E. Popova c, M.A. Srokosz c

a Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College, South Kensington Campus, London SW7 2AZ, UKb Grantham Institute for Climate Change, Imperial College, South Kensington Campus, London SW7 2AZ, UKc National Oceanography Centre, Southampton, University of Southampton Waterfront Campus, European Way, Southampton SO14 3ZH, UK

a r t i c l e i n f o

Article history:Received 26 May 2012Received in revised form 20 July 2012Accepted 22 July 2012Available online 31 July 2012

Keywords:Unstructured meshTurbulence parameterisationAdaptive meshFinite element

1463-5003/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ocemod.2012.07.003

⇑ Corresponding author.E-mail address: [email protected] (J. Hill).

a b s t r a c t

Research into the use of unstructured mesh methods for ocean modelling has been growing steadily inthe last few years. One advantage of using unstructured meshes is that one can concentrate resolutionwhere it is needed. In addition, dynamic adaptive mesh optimisation (DAMO) strategies allow resolutionto be concentrated when this is required. Despite the advantage that DAMO gives in terms of improvingthe spatial resolution where and when required, small-scale turbulence in the oceans still requiresparameterisation. A two-equation, generic length scale (GLS) turbulence model (one equation for turbu-lent kinetic energy and another for a generic turbulence length-scale quantity) adds this parameterisa-tion and can be used in conjunction with adaptive mesh techniques. In this paper, an implementationof the GLS turbulence parameterisation is detailed in a non-hydrostatic, finite-element, unstructuredmesh ocean model, Fluidity-ICOM. The implementation is validated by comparing to both a labora-tory-scale experiment and real-world observations, on both fixed and adaptive meshes. The model per-forms well, matching laboratory and observed data, with resolution being adjusted as necessary byDAMO. Flexibility in the prognostic fields used to construct the error metric used in DAMO is requiredto ensure best performance. Moreover, the adaptive mesh models perform as well as fixed mesh modelsin terms of root mean square error to observation or theoretical mixed layer depths, but uses fewer ele-ments and hence has a reduced computational cost.

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1. Introduction

New numerical methods based upon unstructured meshes, andused in conjunction with anisotropic mesh optimisation methods,have the potential to become an important tool for simulatingocean dynamics. Modelling the oceans accurately demands meth-ods that can represent highly complex domains, capture multi-scale and anisotropic dynamics, and replicate the strong couplingacross a wide range of scales present in the system. An examplewhere it is advantageous to be able to resolve sub-mesoscale phe-nomena whilst simulating a basin-scale domain in an ocean oligo-trophic gyre. They are characterised by low rates of primaryproduction but their great area, covering roughly a third of theEarth’s surface, and probably constituting the largest ecosystemon the planet, means that they play a crucial role in global biogeo-chemistry. Current basin-scale models give values of primary pro-duction approximately two orders of magnitude lower than thoseobserved, thought mainly to be due to the failure in resolvingsub-mesoscale phenomena, which may play a significant role in

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nutrient supply in such areas (Klein and Lapeyre, 2009). However,which aspects of sub-mesoscale processes are responsible for theobserved higher productivity is an open question (McGillicuddyet al., 2003; Lévy et al., 2012). Existing models are limited by twoopposing requirements: to have high enough spatial resolution toresolve fully the processes involved (down to order 1 km) andthe need to realistically simulate the full gyre. No model can cur-rently satisfy both of these constraints.

The Imperial College Ocean Model (Fluidity-ICOM) is a fullyhydrodynamic, non-hydrostatic, finite-element ocean model whichuses unstructured tetrahedral mesh technology in an attempt tomaximise accuracy and computational efficiency (Pain et al.,2005, 2001; Piggott et al., 2005; Ford et al., 2004). The use ofunstructured grids has several advantages including accurate rep-resentation of coastlines and bathymetry, and refinement of thegrid in areas that require it such as embayments or straits (Painet al., 2005). Additionally, Fluidity-ICOM is able to make use of dy-namic adaptive mesh optimisation (DAMO) (Piggott et al., 2005,2009) that is capable of refining the mesh in certain regions andat certain times in order to focus the available computational re-source and lead to greater accuracy, while coarsening of the meshoccurs where fewer elements are required to maintain accuracy

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2 J. Hill et al. / Ocean Modelling 56 (2012) 1–15

according to some user-defined metric. While DAMO is often ap-plied fully in three dimensions or just in the horizontal, it maybe preferable to use adaptive mesh techniques in two orthogonaldirections, horizontally and vertically, independent of each otherin large-scale ocean simulations. Fluidity-ICOM’s DAMO is usedfor a number of applications areas, including biological models,such as NPZD (Nutrient-Phytoplankton-Zooplankton-Detritus).Such biological models require that the upper ocean physics besimulated accurately, including correct simulation of the uppermixed layer (Popova et al., 2006). However, in order to simulatethe upper mixed layer correctly a turbulence parameterisationscheme is required.

Despite the adaptive mesh capabilities of Fluidity-ICOM, a reli-able parameterisation of the eddy diffusivity and viscosity coeffi-cients is still required to accurately represent vertical turbulentfluxes. Most ocean models use a one- or two-equation scheme,with the Mellor–Yamada level 2.5 (Mellor and Yamada, 1982)and the k� � (Rodi, 1987) models remaining popular. Recentdevelopments have seen the implementation of the generic lengthscale (GLS) turbulence parameterisation (Umlauf and Burchard,2003), which can be set-up to give equivalent formulations to pop-ular models by merely changing a few parameters. This gives theGLS model a distinct advantage within an ocean model, such asFluidity-ICOM, as it allows an identical numerical implementationof several turbulence parameterisations. Given the complexity andtime associated with implementing a turbulence model and havingto take into account the changing mesh resolution, we have imple-mented the GLS turbulence parameterisation within Fluidity-ICOM. Here, we describe the implementation of four two-equationturbulence models, within the GLS framework. The approach takenis similar to that of Warner et al. (2005) who implemented thesame parameterisations within a finite-difference ocean model,ROMS. Our implementation will allow the simulation of a widenumber of process scales using both refined meshes and turbu-lence parameterisation.

Previous attempts have been made at using mesh movementtechniques with a turbulence parameterisation scheme (Burchardand Beckers, 2004), which give a much lower error compared tofixed coordinate simulations. This techniques was then extendedto a finite-element code, but restricted to a Mellor–Yamada model(Hanert et al., 2006) and then to three-dimensions (Hofmeisteret al., 2010). However, all previous implementation are such thatadaptivity occurs in the vertical only and the number of layerswas fixed a priori in all simulations. The objective of the adaptivemesh technique was therefore a reduction in error. Adding adap-tivity as per previous studies increases computational cost by asmuch as 40% (Hofmeister et al., 2010). In contrast, the dynamicadaptive mesh optimisation techniques in Fluidity-ICOM allowthe number of elements to vary throughout the simulation. Forexample, it may be beneficial to have different numbers of layersfor different seasons. Not only can DAMO give lower error mea-sures as with previous studies, but also decrease total computa-tional time due to the lower number of elements required andalso eliminating the requirement of fixing the number of verticallayers in a simulation a priori.

This article outlines the general Fluidity-ICOM formulation, be-fore describing the implementation of the GLS framework. We thendescribe the adaptive algorithms within Fluidity-ICOM. The GLSmodel is then validated against laboratory-scale experiments ofKato and Phillips (1969), where we also perform a direct compar-ison against the performance of the General Ocean TurbulenceModel, GOTM (Burchard and Petersen, 1999), with both fixed andadaptive meshes. Finally, we show that Fluidity-ICOM can success-fully model a decade of upper mixed layer depth measurementsforced by real atmospheric fluxes at a range of vertical resolutions,including using an adaptive mesh in the vertical direction.

2. Model formulation

2.1. Fluidity-ICOM

Fluidity-ICOM is a highly flexible finite element/control volumemodelling framework which allows for the numerical solution of anumber of equation sets. Here the non-hydrostatic Boussinesqequation system is considered

@u@tþ u � ruþ f k� n� u ¼ �r p

q0

� �� q

q0gkþr � mruð Þ; ð1aÞ

r � u ¼ 0; ð1bÞ@T@tþ u � rT ¼ r � jTrTð Þ; ð1cÞ

@S@tþ u � rS ¼ r � jSrSð Þ; ð1dÞ

q � qðT; SÞ; ð1eÞ

where u is the 3D velocity vector, t represents time, p is the pres-sure, g is the acceleration due to gravity acting in the k ¼ ð0;0;1Þs

direction, T is temperature and S is salinity. q is the density whichis given in terms of an equation of state function with temperatureand salinity as input arguments, and q0 is a constant backgroundvalue for density. m is the tensor of kinematic viscosities andjT ;jS are the thermal and saline diffusivity tensors respectively. fis the Coriolis parameter which in this work is assumed constant.We also assume, for simplicity, a Cartesian coordinate system withk pointing in the direction of gravity.

The above equations are discretised on an unstructured mesh oftetrahedral elements using the finite element method where allsolution variables are represented with piecewise polynomials.The order of these polynomials for the different solution variables,and whether or not they are continuous or discontinuous acrosselement faces, dictates many of the numerical properties of the dis-cretisation. Here, we demonstrate the GLS stability on two formu-lations; continuous piecewise linear and control volumes for thescalar fields of the GLS formulation.

In time a two-level h method is employed, here h ¼ 0:5 whichyields an implicit Crank-Nicolson scheme. In order to segregatethe solution procedures for velocity and pressure a projectionmethod is employed. Two Picard iterations per time step are usedto linearise the nonlinear advection term. Further details of the dis-cretisation methods employed are given in Piggott et al. (2008) andPain et al. (2010).

2.2. Boundary conditions

The domain used for both sets of experiments in this study wasthree-dimensional. The boundary conditions that are common toboth experiments are as follows. For velocity, ‘‘no normal flow’’boundary conditions are applied at the top and bottom of the do-main, with stress-free tangential conditions at the bottom of thedomain and an applied wind stress at the top surface. In theKato–Philips test (section 3.1) this was a prescribed wind stress.For ocean weather station P (section 3.2), the wind stress was sup-plied by using bulk formulæ(Section 2.13). More details on theboundary conditions for each experiment are given in the relevantsections.

2.3. GLS Turbulence model

Four two-equation turbulence models have been implementedin Fluidity-ICOM, based upon the descriptions given in Umlaufand Burchard (2003) and Warner et al. (2005). Briefly, the modelrelies on a local, temporally varying, kinematic vertical eddy

Table 1Generic length scale parameters.

Model: k� kl k� � k�x genW ¼ k1l1 c0

l

� �3k

32l1 c0

l

� ��1k

12l1 c0

l

� �2k1l

23

p 0.0 3.0 �1.0 2.0m 1.0 1.5 0.5 1.0n 1.0 �1.0 �1.0 �0.67rk 2.44 1.0 2.0 0.8rW 2.44 1.3 2.0 1.07c1 0.9 1.44 0.555 1.0c2 0.5 1.92 0.833 1.22cþ3 1.0 1.0 1.0 1.0kmin 5:0� 10�6 7:6� 10�6 7:6� 10�6 7:6� 10�6

Wmin 1:0� 10�8 1:0� 10�12 1:0� 10�12 1:0� 10�12

Fwall See Section 2.4 1.0 1.0 1.0

Table 2Values for the c�3 parameter for each combination of closure scheme and stabilityfunction

Model: k� kl k� � k�x genW ¼ k1l1 c0

l

� �3k

32l1 c0

l

� ��1k

12l1 c0

l

� �2k1l

23

KC 2.53 �0.41 �0.58 0.1CA 2.68 �0.63 �0.64 0.05CB – �0.57 �0.61 0.08GL – �0.37 �0.492 0.1704

J. Hill et al. / Ocean Modelling 56 (2012) 1–15 3

viscosity, mM , and vertical eddy diffusivity, mH , that parametrisesturbulence (the local Reynolds stress, uiuj

1 in tensoral form) as:

u0w0 ¼ �mM@u@z; v 0w0 ¼ �mM

@v@z; w0q0 ¼ �mH

@q@z

; ð2Þ

with

mM ¼ k12lSM þ m0

M; mH ¼ k12lSH þ m0

H: ð3Þ

Here, we follow the notation of Umlauf and Burchard (2003),where u and v are the horizontal components of the Reynolds-averaged velocity along the x- and y-axes, w is the vertical velocityalong the vertical z-axis, positive upwards, and u0;v 0 and w0 are thecomponents of the turbulent fluctuations about the mean velocity.m0

H is a background value diffusivity, m0M is a background value for

viscosity, SM and SH are often referred to as stability functions, kis the turbulent kinetic energy, and l is a length-scale. SM and SH

also contain a number of constants which are dependent on themodel and stability function used. When using GLS the values ofmM and mH are added to the background values, m0

M and m0H respec-

tively, to become the vertical components of m and jT in Eq. (1e).Other tracer fields, such as salinity use the same diffusivity as tem-perature, i.e. jT ¼ jS.

The generic length scale turbulence closure model (Umlauf andBurchard, 2003) is based on two equations, for the transport of tur-bulent kinetic energy (TKE) and a generic second quantity, W. TheTKE equation is:

@k@tþ ui

@k@xi¼ @

@zmM

rk

@k@z

� �þ P þ B� �; ð4Þ

where rk is the turbulence Schmidt number for k;ui are the velocitycomponents, and P and B represent the turbulent kinetic energyproduction through shear and buoyancy effects, and are defined as:

P ¼ �u0w0@u@z� v 0w0 @v

@z¼ mMM2; M2 ¼ @u

@z

� �2

þ @v@z

� �2

; ð5Þ

B ¼ � gq0

q0w0 ¼ �mHN2; N2 ¼ � gq0

@q@z

: ð6Þ

Here N is the buoyancy frequency, M is the vertical velocityshear and q0 is a density constant which when using the Bous-sinesq approximation is 1. The dissipation is modelled using a rateof dissipation term:

� ¼ c0l

� �3þpnk

32þ

mn W�

1n ð7Þ

where c0l is a model constant which can be used to make W identi-

fiable with any of the more familiar turbulence models, e.g. kl; �, andx, by adopting the values shown in Table 1 (Umlauf and Burchard,2003). The second equation is:

@W@tþ ui

@W@xi¼ @

@zmM

rW

@W@z

� �þW

kðc1P þ c3B� c2�FwallÞ: ð8Þ

The parameter rW is the Schmidt number for W and ci are con-stants based on experimental data. The value of c3 depends onwhether the flow is stably stratified (in which case c3 ¼ c�3 ) orunstable (c3 ¼ cþ3 ). Values for c�3 for all possible combinations ofstability functions and turbulence models included in Fluidity-ICOM are shown in Table 2. Here,

W ¼ c0l

� �pkmln ð9Þ

and

l ¼ c0l

� �3k

32��1: ð10Þ

1 We use u1 ¼ u;u2 ¼ v , and u3 ¼ w.

By choosing values for the parameters p;m;n;rk;rW; c1; c2; c3,and c0

l one can recover the exact formulation of three standardGLS models, k� �; k� kl; k�x, and an additional model basedon Umlauf and Burchard (2003), the gen model (see Table 1 forvalues).

2.4. Wall functions

The k� kl closure scheme (see Section 2.8) requires the use of awall function to ensure correct behaviour in the viscous sublayer asthe value of n is positive (see Umlauf and Burchard, 2003). All otherschemes do not require a wall function. There are four differentwall functions that have been implemented within Fluidity-ICOM.In standard Mellor–Yamada models (Mellor and Yamada, 1982),Fwall is defined as:

Fwall ¼ 1þ E2lj

db þ ds

dbds

� �2 !

; ð11Þ

where E2 ¼ 1:33, and ds and db are the distance to the surface andbottom respectively.

An alternative suggestion by Burchard et al. (1998) suggests asymmetric linear shape:

Fwall ¼ 1þ E2lj

1MIN db;dsð Þ

� �2 !

: ð12Þ

Burchard and Bolding (2001) used numerical experiments todefine a wall function simulating an infinitely deep basin:

Fwall ¼ 1þ E2lj

1ds

� �2 !

: ð13Þ

Finally, Blumberg et al. (1992) suggested a correction to thewall function for open channel flow:

Fwall ¼ 1þ E2l

jdb

� �2

þ E4l

jds

� �2 !

; ð14Þ


where E4 ¼ 0:25.

2.5. k� � model

The k� � model is based on that derived by Rodi (1987), whichcan be obtained from the GLS model by setting p ¼ 3:0; m ¼ 1:5,and n ¼ �1:0.

@k@tþ ui

@k@xi¼ @

@zmM

rk

@k@z

� �þ P þ B� � ð15Þ

and

@�@tþ ui

@�@xi¼ @

@zmM

r�@�@z

� �þ �

kðc1P þ c3B� c2�Þ; ð16Þ

where rk (1.0) is the Schmidt number for the eddy diffusivity ofturbulent kinetic energy and r� (1.3) is the Schmidt number for theeddy diffusivity of dissipation with c�1 ¼ 1:44; c�2 ¼ 1:92, and c�3takes a value depending on the stability function used.

2.6. k�x model

The parameter x can be considered ‘a frequency characteristicof the turbulence decay process’ (Wilcox, 1988) and is related todissipation by:

x ¼ �ðc0

lÞ4k: ð17Þ

Setting p ¼ �1:0; m ¼ 0:5, and n ¼ �1:0 gives the two turbu-lence equations:

@k@tþ ui

@k@xi¼ @

@zmM

rxk

@k@z

� �þ P þ B� � ð18Þ

and

@x@tþ ui

@x@xi¼ @

@zmM

rx

@x@z

� �þx


where rxk (2.0) is the Schmidt number for the eddy diffusivity of

turbulent kinetic energy and rx (2.0) is the Schmidt number forthe eddy diffusivity of x.

2.7. gen model

Following Warner et al. (2005), we have also implemented a genmodel based on the arguments of Umlauf and Burchard (2003)(their Table 7). The model is created by specifying p ¼ 2:0; m ¼

1:0, and n ¼ �0:67, yielding W ¼ c0l

� �2kl�2=3 ¼ c,

@k@tþ ui

@k@xi¼ @

@zmM

rck

@k@z

!þ P þ B� � ð20Þ

and

@c@tþ ui

@c@xi¼ @

@zmM

rc

@c@z

� �þ c


where rck ¼ 0:87 and rc ¼ 1:07 are the Schmidt numbers of eddy

diffusivity of turbulent kinetic energy and c respectively.

2.8. k� kl model

The k� kl model is a variant of the Mellor–Yamada-type turbu-lence model (Mellor and Yamada, 1982). To arrive at k� kl, setp ¼ 0; m ¼ 1:0, and n ¼ 0, which yields:

@k@tþ ui

@k@xi¼ @

@zmM

rk

@k@z

� �þ P þ B� � ð22Þ

and

@

@tðklÞ þ ui

@

@xiðklÞ ¼ @

@zmM

rk

@

@zðklÞ

� �þ lðc1P þ c3B� Fwallc2�Þ;

ð23Þ

where rk (2.44) is the Schmidt number for k.

2.9. Boundary conditions

The boundary conditions for the two GLS equations can beeither of Dirichlet or Neumann type. For the turbulent kinetic en-ergy, the Dirichlet condition can be written as:

k ¼ u�ð Þ2

c0l

� �2 ; ð24Þ

where u� is the friction velocity. A corresponding flux condition istherefore:

mM

rk

@k@z¼ 0: ð25Þ

Similarly for the generic quantity, W, the Dirichlet condition iswritten as:

W ¼ c0l

� �plnkm

: ð26Þ

At the top of the viscous sublayer the value of W can bedetermined from Eq. (9), specifying l ¼ jz and k from Eq. (24),gives:

W ¼ c0l

� �p�2mu�s� �2m jzsð Þn; ð27Þ

where zs is the thickness of the viscous sublayer and u�s is the fric-tion velocity at the surface or bottom respectively.

Calculating the corresponding Neumann conditions by differen-tiating with respect to z, yields:

mM

rW

@W@z¼ n

mM

rWc0l

� �pjnkmzn

s : ð28Þ

Note that it is also an option to express the Neumann conditionabove in terms of the friction velocity, u�. Previous work has shownthat this can cause numerical instability in the case of stress-freesurface boundary layers as the friction velocity is near zero (Bur-chard and Petersen, 1999), so is not considered further. Dirichletboundary conditions for the second equation can be unstable fork� � models unless the resolution at the boundary is very high(Burchard and Petersen, 1999). It is therefore advisable to use theNeumann flux boundary conditions (Eqs. (25) and (28)), howeverboth are available as options to the user.

2.10. Stability functions

The second-order moments are now closed, bar the definition ofthe stability functions, SM and SH , which are functions of aM and aN ,defined as:

aM ¼k2

�2 M2; aN ¼k2

�2 N2: ð29Þ

The two stability can be defined as:

SMðaM;aNÞ ¼n0 þ n1aN þ n2aM

d0 þ d1aN þ d2aM þ d3aNaM þ d4a2N þ d5a2

M

ð30Þ

and

SHðaM;aNÞ ¼nb0 þ nb1aN þ nb2aM

d0 þ d1aN þ d2aM þ d3aNaM þ d4a2N þ d5a2

M

: ð31Þ

Table 3Critical Richardson numbers, Ri ¼ Ric , for the four stability functions.

Model: GL78 KC94 C01A C01B

Ric 0.47 0.24 0.85 1.02


The parameters n0;n1;n2;d0;d2;d3; d4;nb0, nb1; nb2 depend on themodel parameters chosen and can be related to traditional stabilityfunctions (Umlauf and Burchard, 2005). However, using the equilib-rium condition for the turbulent kinetic energy as P þ Bð Þ=� ¼ 1, onecan write aM as a function of aN allowing elimination of aM in theabove equations (Umlauf and Burchard, 2005):

SMðaM;aNÞaM � SNðaM ;aNÞaN ¼ 1: ð32Þ

A limit on negative values of aN needs to applied to ensure aM doesnot also become negative.

The GLS model included in Fluidity-ICOM contains four choicesof parameters that form the stability functions. These are: GL78(Gibson and Launder, 1978), KC94 (Kantha and Clayson, 1994),C01A and C01B (Canuto et al., 2001). Using the above assumptionto remove aM , the equilibrium models used here have a criticalRichardson number, Ric , where:

Ri ¼ N2

S2 : ð33Þ

For Ri ¼ Ric the equilibrium models predict complete extinction ofall turbulence. For the four stability function choices included inFluidity-ICOM the critical Richardson number is given in Table 3.

2.11. Length scale limits

An upper limit for l is required to simulate the limiting effect ofstable stratification (Galperin et al., 1988):

l26

0:56k

N2 for N2 > 0: ð34Þ

This limit can be written in terms of W and �:

W1=n Pffiffiffiffiffiffiffiffiffiffi0:56p

k c0l

� �p=nkm=nþ1=2N�1 ð35Þ

and

�Pc0l

3kN2ffiffiffiffiffiffiffiffiffiffi0:56p : ð36Þ

We found that the limits on � and l are required in cases wheren < 1 (all models except the k� kl model) and the W limit is onlyrequired for cases where n > 1 (k� kl model). Like (Warner et al.,2005) these limits are imposed on all calculations.

2.12. Dynamic adaptive mesh optimisation

The mesh adaptivity algorithm used in this work attempts tooptimise the size as well as the shape of individual elements ofthe mesh in order to minimise an optimisation functional (Painet al., 2001; Piggott et al., 2005, 2008). This functional is madeup of terms which encode the size and shape of elements. Sincea metric tensor is used to evaluate the functional, and this metrictensor is free to vary as a function of space this consequently leadsto a mesh which is both inhomogeneous and anisotropic in sizeacross the domain.

In Fluidity-ICOM, mesh adaptivity aims to increase resolution inregions of the domain with large curvatures of given fields and de-crease resolution elsewhere. This approach is motivated by inter-polation error theory and allows good representation of thesmall-scale dynamics and sharp gradients without the need forhigh spatial resolution throughout the entire domain, (Piggott

et al., 2005). The mesh is adapted through a series of local topolog-ical and geometrical operations as described in Pain et al. (2001).These operations seek to produce a mesh where all elements haveunit edge-length when measured with respect to a given metric, M.

In Fluidity-ICOM a relatively simple metric is employed. Forchosen fields, fi, metrics, Mi is given by:

M ¼ det Hj j�1

2pþnHj je; ð37Þ

where e is a user-defined weight corresponding to the field underconsideration and jHj is the Hessian matrix for the field under con-sideration where the absolute values of its eigenvectors have beentaken, p 2 Z and n is the dimension of the space (Loseille and Ala-uzet, 2011). The final metric used, M, is formed from a superpositionof the metrics for individual fields: M ¼

SiMi, (Pain et al., 2001). In

the work presented here we test values of p of 2 and1 as both havebeen used in previous work, but p ¼ 2 has shown superior results inresolving both weak and strong curvatures within the same simula-tion (Loseille and Alauzet, 2011). The use of a tensor for the metricallows anisotropic directional information to be included and influ-ence the adaptivity. � may vary spatially and temporally, but nei-ther is utilised here. In general, for a given solution field,decreasing � will lead to greater refinement of the mesh andincreasing � will lead to more coarsening. The maximum and min-imum edge-length allowed can also be specified by the user. Formore details see Pain et al. (2001), Piggott et al. (2005, 2008) andreferences therein.

The mesh is adapted at run-time through local operations andthe frequency with which it adapts is specified by the user. Afteran adapt the solution fields must be interpolated from the pre-to post-adapt mesh. Two methods are available ‘consistent-inter-polation’ and ‘bounded Galerkin-projection’, but only the formeris considered here. See Farrell et al. (2009) for detailed descriptionsof these two interpolation types. All prognostic fields are interpo-lated using the user-specified method, along with any diagnosticfields the user has specified to be interpolated. For the GLS equa-tion, k and W are interpolated and (unless specified otherwise)all other variables are re-calculated following an adapt.

The adaptive mesh technique used in Fluidity-ICOM differsfrom previous implementations of adaptive mesh techniques usedwith turbulence parameterisations in that the number of elements(or in the case of finite-difference models, grid points) can changethroughout the simulation. Burchard and Beckers (2004) diffusethe grid points based on a grid diffusion coefficient which is com-posed of a number of components: the gradient of density differ-ence, shear velocity difference, near-surface component, andbackground component. An identical method was employed byHanert et al. (2006) using a finite-element model and a Mellor–Yamada turbulence closure schemes. The method of constructingthe error metric used by these studies is very similar to that of Flu-idity-ICOM, but uses gradients rather than curvature. However, thenumber of grid points remains fixed: the adaptive mesh simplymoves them to locations to minimise the error metric; in essencea mesh movement algorithm. The techniques of Burchard and Bec-kers (2004) have been extended to 3D by allowing each horizontallocation to have a different vertical mesh (Hofmeister et al., 2010).Again, the number of grid points is fixed and the overhead whilstadapting the grid is an 30–40% increase in run time. However,adaptive techniques have been shown to reduce the amount ofnumerical mixing in a number of idealised examples (Hofmeisteret al., 2010).

2.13. Ocean forcing

In order to simulate real-world ocean scenarios to test the tur-bulence models, momentum, freshwater and heat fluxes must be

Table 4Values of key model parameters for the Kato–Philips channel run.

Model parameter Value

Number of elements 15000Timestep 30 sMesh size 25 � 5 m in horizontal; 5 m – 0.125 m in verticalOutput Every 3600 sInitial temperature Linear profile from 13�C on surface to 10�C at baseSurface stress 0.1 N m�2

Domain size 250 � 50 � 50 m, periodic in x-direction


applied to the upper surface. We have implemented the bulk for-mulæ of Large and Yeager (2004) in combination with the ERA-40 reanalysis data (Uppala et al., 2005).

There are three surface kinematic fluxes calculated: heat – hwhi,salt – hwsi, and momentum – hwui and hwvi, which can be relatedto the surface fluxes of heat Q, the freshwater F, and the momen-tum s ¼ su; svð Þ, via:

hwhi ¼ Q � qCp� ��1

; ð38Þhwsi ¼ F � q�1S0

� �; ð39Þ

hwui; hwvið Þ ¼ sq�1 ¼ su; svð Þ � q�1; ð40Þ

where q is the ocean density, Cp is the heat capacity (4000 Jkg�1K�1)and S0 is a reference ocean salinity, which is the current sea surfacesalinity. These fluxes are then applied as upper-surface Neumannboundary conditions on the appropriate fields.

3. Model validation

3.1. Kato–Philips

The experiment described by Kato and Phillips (1969) invokes aconstant surface momentum stress applied to an initially quiescentfluid with a uniform density gradient due to a linear temperaturegradient. As time progresses the mixed layer deepens due to thisinduced stress. This is a common test for evaluating the perfor-mance of turbulence models and stability functions (Burchardand Bolding, 2001). Similar to Warner et al. (2005), the model do-main is 250 m long, 50 m wide and 50 m deep, with free-slip con-ditions applied to the tank sides. The two ends of the tanks parallelto the stress direction are periodic, making an infinite channel,50 m wide. The temperature has no flux boundary conditions ap-plied to all surfaces. The mixed layer depth (MLD) is defined asthe depth at which k < 10�5 m2 s�2 as per previous studies (e.g.,Warner et al., 2005). Based on experimental data, Price (1979) gavea correlation of the deepening of the MLD (Dm) with time (t) as:

Dm ¼ 1:05u�s N�1=20 t1=2; ð41Þ

where the surface friction velocity u�s ¼ffiffiffiffiffiffiffiffiffiffiffiffiss=q0

p;N0 is the initial

buoyancy frequency, and takes the value 0:01 s�1 for thissimulation.

3.1.1. Fixed meshThe four turbulence closure schemes (k� kl; k� �; k�x; gen)

were tested with each stability function, giving a total of 14 simu-lations (k� kl was only implemented with KC94 and C01A stabilityfunctions). A further suite of simulations using the k� � closurewith the C01A stability functions were run on a range of verticalresolutions from 5 m to 0.125 m. For further validation we alsoran the same problems using GOTM (Burchard et al., 1999) (exceptthe k� kl models as GOTM (version 4.0) does not include wallfunctions). Both Fluidity-ICOM and GOTM use a timestep of 30 s,but Fluidity-ICOM uses a ‘‘spin-up’’ timestep where the timestepincreases from 10 s to 30 over six timesteps. This was to avoid ini-tial spurious changes to the GLS prognostic fields. The simulationswere run with both P1P1 velocity–pressure element pair with trac-ers (including GLS) using a P1CG discretisation, and P1P1 velocity–pressure element pair with tracers (including GLS) using a P1CV dis-cretisation. Other model parameters are given in Table 4.

Results for the mixed layer depth evolution are shown in Fig. 1and Table 5. All model and stability function combinations showgood agreement with the predicted mixed layer behaviour of Price(1979). Of the four models implemented, k� � and k�x give amixed layer depth that is the most accurate (Fig. 1). A standard

Root Mean Square error (RMS) was then calculated to the mixedlayer depth predicted by Eq. (41) according to:

error ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPxi � yið Þ2

n

s; ð42Þ

where xi is then predicted mixed layer depth (e.g. Eq. (41)) and yi isthe mixed layer depth produced by the simulation being consid-ered. Both GOTM and Fluidity-ICOM show similar RMS error toPrice (1979), ranging from 1.44 m to 0.112 m (Table 5). Fluidity-ICOM gives errors well within the range of GOTM and hence weconclude that Fluidity-ICOM predicts the mixed layer depth behav-iour well in this test case. We also compared the profiles of k and mM

against the same profiles for GOTM (Fig. 2). Apart from minor differ-ences in the maximum value of mM , the only difference is the minorvariation at the upper boundary. This is due to differences in howthe boundary conditions are handled between the finite-differencebased GOTM (the boundary condition is not applied on the topolog-ical surface, but within the computational grid) and the finite-ele-ment based Fluidity-ICOM (the boundary condition is applied atthe topological surface).

To further test Fluidity-ICOM’s behaviour in this test case, weperformed the same simulation, but at different vertical resolu-tions. By increasing vertical resolution the simulation should con-vergence onto the ‘‘correct solution’’. As no true analytical solutionis available we consider comparisons to both Price (1979) and thehighest resolution simulation (0.125 m) for the equivalent modelor discretisation. Results from this analysis show approximatelylinear convergence rate using both CV and CG discretisations to-wards the highest resolution simulation (Tables 6 and 7 andFig. 3). GOTM shows similar behaviour. The lowest error to Price(1979) was 0.43 m (0.468 m for GOTM), but this occurred at 1 mvertical resolution for Fluidity-ICOM and 0.5 m for GOTM. Thereis little difference between CG and CV discretisations, though CGshows lower error at lower resolutions.

3.1.2. Dynamically adapted meshTo assess the effect of adaptivity on GLS we used the 3D adap-

tivity algorithm described in Section 2.12, resulting in a fullyunstructured mesh – that is, unstructured in all three dimensions.All adaptive tests were carried out using the k� � settings for theGLS model and the C01A stability functions, although other mod-el/stability function pairs are expected to behave similarly. A num-ber of error metrics were tried with varying interpolation errorsspecified. Results are shown for a subset of all metrics assessed.Metrics presented here all used two fields in a paired combination:v � q; k�W, and N2 �M2. We use these pairs as they indirectlyrepresent the buoyancy and stress sources for the turbulenceparameterisation and therefore represent a sensible choice onwhich to control resolution of the mesh. This is a similar strategyto previous studies (Burchard and Beckers, 2004; Hanert et al.,2006; Hofmeister et al., 2010). The values of e (Eq. (37)) for thethree field-pairs were: (v ¼ ½1� 10�3;1;1�; q ¼ 0:001), (k ¼ 1�10�5; W ¼ 1� 10�8), and (N2 ¼ 1� 10�4; M2 ¼ 1� 10�5) respec-

Fig. 1. Mixed layer depth from the Kato–Philips channel test with fixed meshes for the fourteen embedded models-stability function combinations in Fluidity-ICOM alongwith the predicted MLD of Price (1979).

Table 5RMS error to Price (1979) for each GLS modelvariant available in Fluidity-ICOM using CV dis-cretisation and 1 m vertical resolution. Equivalenterror for GOTM is shown in parentheses.

Model Error to Price (1979) (m)

gen-CA 0.878 (1.440)gen-CB 0.855 (0.952)gen-KC 0.218 (0.112)gen-GL 1.076 (0.454)k� �-CA 0.210 (0.812)k� �-CB 0.288 (0.952)k� �-KC 0.281 (0.112)k� �-GL 0.526 (0.455)k�x-CA 0.201 (0.152)k�x-CB 0.194 (0.983)k�x-KC 0.673 (0.280)k�x-GL 0.354 (0.347)k� kl-CA 0.538 (–)k� kl-KC 0.240 (–)


tively In addition we experimented with setting p in Eq. (37) to 2and 1 (labelled L2 and L1). Adaptivity was performedevery ten timesteps, though simulations (not shown) were also

performed adapting every 20, 30 and 50 timesteps. All showedsimilar behaviour, but will vary in their computational cost. Allother settings were as described in Table 4.

The adaptivity performed as expected, increasing the verticalresolution in the mixed layer, whilst reducing it below, giving asteady increase in the number of elements as the simulation pro-ceeds (Fig. 5). The adaptivity algorithm also placed increased reso-lution at the bottom surface of the channel in response to theboundary conditions of both k and W when using these fields toform the adaptivity metric. A final note on the adaptive mesh isthe anisotropy of the mesh with high vertical resolution and sub-stantially lower horizontal resolution. This is in part due to theminimum and maximum edge lengths specified, but also becausethe error metric was constructed from the fields that show mostchange in the vertical direction. Consequently, adaptivity will con-centrate resolution in the vertical direction. As the mixed layerdeepens, the number of nodes in the simulation increases, trackingthe mixed layer. As the resolution increases near the surface thevalue of k approaches that of the Dirichlet boundary condition im-posed by GOTM (Fig. 2).

To test the effect of the dynamic adaptive mesh optimisation wecompare the results of the adaptive run to a series of fixed meshsimulations. The results (Table 8 and Fig. 4) show that adaptivity

Fig. 2. Vertical profiles of k (left panel) and mM (middle panel) and mixed layer depth behaviour for k� � (top-left), gen (top-right), k�x (bottom-left), k� � on an adaptivemesh (bottom-right) for Fluidity-ICOM (solid line). The equivalent vertical profiles for GOTM are also shown (dotted line). For more information on adaptive mesh behavioursee Section 3.1.2.

Table 6Results of the convergence test for the Kato–Philips channel test for GLS on a ControlVolume discretisation. Results from GOTM compared to the 0.125 m resolution GOTMsimulation or Price (1979) are in parenthesis.

Vertical Resolution (m) Error to 0.125 m (m) Error to Price (1979) (m)

5 3.183 (4.168) 2.674 (3.829)4 2.327 (3.549) 1.779 (3.187)2 1.030 (1.632) 0.540 (1.396)1 0.285 (0.759) 0.430 (0.619)0.5 0.211 (0.307) 0.548 (0.468)0.25 0.128 (0.099) 0.580 (0.534)0.125 – 0.579 (0.574)

Table 7Results of the convergence test for the Kato–Philips channel test for GLS on aContinuous Galerkin discretisation. Results from GOTM compared to the 0.125 mresolution GOTM simulation or Price (1979) are in parenthesis.

Vertical Resolution (m) Error to 0.125 m Error to Price (1979)

5 2.227 (4.168) 1.793 (3.829)4 1.719 (3.549) 1.315 (3.187)2 0.937 (1.632) 0.593 (1.396)1 0.274 (0.759) 0.402 (0.619)0.5 0.105 (0.307) 0.492 (0.468)0.25 0.062 (0.099) 0.515 (0.534)0.125 – 0.484 (0.574)


is robust regardless of the error metric used. The best performingmetric (in terms of both reducing the average number of elementsand the smallest RMS) is the metric constructed using k and W.Using the L2 norm rather than L1 gives superior and more robustresults (numerical oscillations, not shown, occurred on some sim-ulations using L1). However, the cost of this is an increase in the

number of elements and vertices in the mesh. The best performingadaptive simulation gives a smaller RMS error compared to Price(1979) than any fixed mesh simulation. However, note that thefixed mesh simulations did not convergence toward Price (1979)(see Fig. 4 and Table 8), but the adaptive simulations do appearto do so as decreasing the minimum edge length reduces theRMS error to the predicted mixed layer depth of Price (1979). Price(1979) is not a fully analytical solution to the GLS equation so can-not be used as the convergence criterion, but the results do showthat both fixed mesh and adaptive mesh simulation perform well,with RMS errors of less than 0.5 m to the predicted mixed layerdepth over a 30 h simulation. Adaptivity gives a lower error anda reduced number of elements, decreasing computational cost.

3.2. Ocean weather station P

Multi-annual simulations using realistic forcing data provide awide range of surface forcing conditions to create a thorough testof an oceanic model (Gaspar et al., 1990). Here, we limit ourselvesto a pseudo-1D test at ocean weather station P (50�N, 145�W), inthe northern Pacific. This location is ideal for testing such 1D mod-els as there are only weak effects from regional advection and ithas been used to test numerous turbulence model closure schemes(e.g., Denman and Miyake, 1973; Gaspar et al., 1990). The test con-sists of a 10 year simulation from 1970 to 1980, using a 1000 mdeep water column. The column is 100 m wide with two nodesin both horizontal directions, i.e. a square horizontal extent splitinto two triangular elements. The initial conditions of the test arethe observed temperature and salinity from the World Ocean Data-base (Boyer et al., 2006). Forcing is implemented as describedabove, which produces a wind-stress boundary condition for theupper surface of the velocity field, and flux conditions for both

Fig. 3. Mixed layer depth behaviour of Fluidity-ICOM at different vertical resolutions (5 m to 0.125 m) using both CG (upper panel) and CV (lower panel) discretisations. Bothshow improved performance to the highest resolution run with increasing resolution. See Tables 6 and 7 for further details.


temperature and salinity. The lower boundary is a Dirichlet condi-tion for all fields. Temperature and salinity are set to the mean an-nual values from the World Ocean Database (Boyer et al., 2006),while velocity is set to a no-flow boundary condition. The lateralboundary conditions for all fields are open boundaries. There isno relaxation to observed data. There are six fixed mesh modelruns in total: using Fluidity-ICOM with a fixed mesh of 1, 2, 5, 10and 20 m, a mesh with depth-dependant resolution (with verticalresolution varying from 3 to 90.8 m with increasing depth withlayer thicknesses based on the UK NEMO model); a number of

simulations using DAMO on a number of metrics and with differentminimum edge lengths; and using GOTM with vertical resolutionsof 1, 2, 5, 10, and 20 m. All simulations used the same starting dataand ERA40-reanalysis data for constructing the surface fluxes.GOTM comes with bulk formula as derived by Kondo (1975). Othermodel parameters are given in Table 9. We used the ‘‘ows’’ testcase available with GOTM (version 4.0) as a basis for this experi-ment, but with the fluxes changed to those derived from ERA40.The example relaxes salinity to measured profiles once per day(every 86400 s). We kept this option for the simulation performed

Table 8RMS error for a variety simulations using DAMO compared to the predicted mixed layer depth of (Price, 1979). The field pair from which the metric was formed, minimum edgelength in the vertical direction and p are shown, along with the same information for the highest resolution (0.125 m) fixed mesh simulation.

Field pair Min. edge length (m) p RMS Error (m) Elements (av., min., max.) Vertices (av., min., max.)

v � q 1.0 1 0.5843 37345.1, 12743, 75000 7950.8, 2825, 15756

N2 �M2 1.0 1 2.5823 24344.0, 12156, 75000 5214.3, 2717, 15756

k�W 1.0 1 1.4088 24612.3, 12257, 75000 5269.0, 2714, 15756v � q 1.0 2 0.4499 50746.9, 21621, 75000 10744.5, 4648, 15756

N2 �M2 1.0 2 0.8529 43840.8, 18300, 75000 9327.9, 3994, 15756

k�W 1.0 2 0.3996 43554.8, 19104, 75000 9247.7, 4139, 15756k�W 0.5 2 0.1643 73662.4, 31653, 94111 15360.8, 6451, 31356Fixed mesh 0.125 – 0.5785 300000 62556


for this work as without this GOTM’s salinity tended to drift awayfrom observed valued. However, no such relaxation was necessaryor used in the Fluidity-ICOM simulations.

Fig. 5. Element count for all adaptive simulations of the Kato–Phillips channel. For all methough for v � q this quickly reaches the maximum as defined by the minimum edge le

Fig. 4. Mixed layer depth behaviour of Fluidity-ICOM using an adaptive mesh.Simulation results are show using three different metrics (calculated usingv � q; k�W, and N2 �M2 as input fields) and for k�W using two differentminimum edge lengths in the vertical (1 m and 0.5 m). All metrics show goodagreement to Price (1979). See Table 8 for further details.

Adaptivity was tested using metrics formed from the same fieldpairs as the Kato–Phillips test, Section 3.1.2. The values of e (Eq.(37)) for the three field-pairs were: (v ¼ ½0:001;0:001;0:1�;q ¼ 1� 10�4Þ; ðk ¼ 1� 10�5; W ¼ 1� 10�7), and (N2 ¼ 1� 10�3;

M2 ¼ 1� 10�4) respectively. For all simulations here we used a va-lue of p ¼ 2 (Eq. (37)) based on the improvement in performanceshown in the Kato–Phillips example. The minimum edge lengthsin the vertical were 2 or 4 m and the maximum was 90 m. Adaptswere performed every 5 h to ensure that the changes in bulk for-mulæ fluxes which had a frequency of 6 h were captured. All valueswere output to disk every 24 h.

The models were compared to observed upper mixed layer(UML) depths, as calculated by Kleypas and Doney (2001), alongwith temperature profiles derived from the World Ocean Database.The base of the UML is defined as the depth at which density was0.125 kg m�3 higher than the surface density, as recommended byKleypas and Doney (2001). All model runs matched well to mea-sured data, simulating the basic behaviour of the UML depththroughout the ten years simulated (Fig. 6). There are years wherethe adaptive Fluidity-ICOM simulation appears to better than thefixed mesh simulation (Fig. 6b). However, there are years whereGOTM simulates a UML closer to that measured, which given therelaxation to measured salinity is not particularly surprising(Fig. 6c). All models also show some short-scale temporal oscilla-tion of the UML depth in the spring, though the relatively low out-put resolution of one day prevents many of these oscillationsappearing in the final output.

trics the number of elements required increases and the mixed layer depth deepens,ngth.

Table 9Values of key model parameters for the ocean weather station P simulations.

Model parameter Value

Timestep 900 sApproximate mesh

size100 � 100 m in horizontal; 1–90.8 m in vertical oradaptive

Output Every 86400 sAdapt frequency Every 18000 s (20 timesteps)Initial temperature From Boyer et al. (2006) (01/01/1970)Initial salinity From Boyer et al. (2006) (01/01/1970)Domain size 100 � 100 � 1000 mElement type Linear Continuous Galerkin for all fields.GLS scheme k� �Stability function C01A (Canuto et al., 2001)


The temperature profiles were also simulated well (Fig. 7), withboth models representing the annual asymmetrical change in tem-perature. Note that GOTM produces a much smoother thermoclinethan that of Fluidity-ICOM and that from observations. GOTM alsoconsistently simulated a sea surface temperatures (SST) higherthan observed. Fluidity-ICOM produced too high SST during thesummer months by 1–2 degrees, whereas GOTM was a further de-gree higher. Note that the adaptive Fluidity-ICOM simulation dif-fers very little from the fixed mesh simulation (Fig. 7).

The adaptive mesh simulations gave good representations ofthe mixed layer depths behaviour, but using fewer elements than

Fig. 6. Ten year plot of Mixed Layer Depth for Fluidity-ICOM (1 m resolution, light greylength using v � q – see Table 10 and text for more information – medium grey line), GBoth models simulate the behaviour of the MLD well. Note that output was every 24 h, hesimulation output, but may have been simulated. GOTM simulations show slightly betterhalf of the simulation (c).

the minimum edge length would otherwise give (Table 10 andFig. 8). Unlike the Kato–Phillips test, the metric using v and q per-formed best, giving the lowest RMS error compared to the 1 mfixed mesh simulation and fewer elements (averaged over the sim-ulation) than the simulation using k and W (287.9 compared to326.3 elements). Previous studies have used N2 and M2 to forman error metric and whilst this simulation required fewest ele-ments on average (165.1), the performance was poor (though notnoticeably incorrect). All adaptive simulations show a seasonalvariation in the number of elements throughout the simulationas the mesh tracks the seasonal variation in mixed layer depth(Fig. 9).

DAMO places extra resolution at the upper surface and the baseof the UML (Fig. 10). For the first 4.5 years of simulated time, thereis also a layer at 150 m depth which contains higher resolution.This is due to the initial conditions describing a sharp gradient inocean density. This in turn leads the adaptive algorithm to placemore resolution here due to the higher curvature. However, as thislayer is diffused, the gradient lessens and eventually the adaptivealgorithm can decrease the resolution here too.

In order to assess the effectiveness of the dynamic adaptivemesh optimisation a convergence test was used. The same simula-tion with different vertical resolutions varying from 20 to 1 m werecompleted. As with the Kato–Philips experiment the error usedwas the RMS error constructed was the mixed layer depth (seeEq. (42)) between the highest resolution run and the simulation

line), adaptive mesh simulation using Fluidity-ICOM (2 m minimum vertical edge-OTM (black line) and observations (black squares) from Kleypas and Doney (2001).nce some ephemeral changes in MLD captured by observations are not present in theMLD behaviour in the first few years (b), but show too much deepening in the latter

Fig. 7. Vertical temperature profiles through time for Fluidity-ICOM with 1 m vertical resolution (a), Fluidity-ICOM with adaptive vertical resolution (2 m minimum verticaledge-length using v � q – see Table 10 and text for more information), GOTM with 1 m vertical resolution (c) and observed profiles (d).

Fig. 8. Mixed layer depths behaviour of all adaptive simulations used in this study and the 1 m fixed resolution Fluidity-ICOM simulation. All show very similar behaviour,but using N2 �M2 shows too much deepening in winter by around 50 m in some years (1974–1976).

Table 10Results of the convergence test for the Ocean Station Papa experiments. The number of elements and nodes are the average, minimum and maximum for adaptive simulations.The NEMO simulation was a Fluidity-ICOM simulation using layer thicknesses varying from 3 m at the surface to 90 m at depth. Equivalent error measurement for GOTM is shownin parentheses.

Resolution Metric Error (1 m simulation) Error (obs.) Vertices (av., min., max.) Elements (av., min., max.)

NEMO – 18.22 (–) 23.13 (–) 148 21620 – 12.85 (15.93) 22.36 (29.32) 204 30010 – 9.74 (9.83) 24.23 (26.11) 504 6005 – 7.06 (6.94) 21.98 (25.00) 804 12002 – 3.01 (2.47) 22.94 (24.88) 2004 30001 – – (–) 23.76 (24.88) 4004 6000Adaptive, 4 m k�W 18.77 (–) 28.49 (–) 326.3, 161, 504 483.5, 236, 752Adaptive, 4 m N2 �M2 24.48 (–) 22.42 (–) 165.1, 52, 265 241.7, 72, 391

Adaptive, 4 m v � q 13.33 (–) 22.38 (–) 287.9, 190, 400 425.8, 280, 594Adaptive, 2 m v � q 11.79 (–) 23.13 (–) 377.8, 233, 553 560.7, 344, 824


Fig. 9. The number of elements used throughout the 10 year Station Papa simulations for all the adaptivity metrics used in this study. All show a seasonal variation as themesh tracks the mixed layer up and down the water column.

Fig. 10. Approximate vertical resolution (Dz) of the Fluidity-ICOM adaptive run at Ocean Station Papa (2 m minimum vertical edge-length using v � q – see Table 10 and textfor more information). The white line shows the position of the UML through time. Adaptivity places extra resolution at the surface and just below the UML as observed in theKato–Philips test. Below the UML the resolution increases to the maximum value of 87 m allowed by the adaptive algorithm. Note the high resolution layer at 150 m depththat lasts for around 4–4.5 years. This is caused by a sharp gradient in the initial density conditions, which slowly diffuses out. Until the error metric composed on theperturbation density can be satisfied by a larger grid, this area must have a high resolution to maintain this gradient. After the summer of 1975, the gradient is no longer sharpenough to require a large number of elements, and the adaptive algorithm removes them.


being considered. As with the Kato–Philips experiment the conver-gence is 1st order (Table 10). The adaptive simulations gaveacceptable results which were within the range of errors producedby GOTM and the fixed-mesh Fluidity-ICOM results and requiredfewer elements, but the error was not substantially lower, as seenin the Kato–Philips test. However, as shown in the Kato–Philipstest it is not necessarily the case that the adaptive run should con-verge towards the highest resolution fixed-mesh simulation.

4. Discussion

In implementing the GLS turbulence parameterisation in a finiteelement model, we came across a number of minor issues that wefeel are worthy of documentation. First, the two GLS prognosticfields, k and W were sensitive to sudden changes in external fields,

both temporally and spatially. These changes could be caused by achange in external forcing (for example at the start of a simulation,where the fluid is at rest and a momentum stress is applied) or by achange in the mesh configuration. Both problems are tractable andsolved by using a small timestep at the start of the simulation orimmediately following an mesh adapt that can increase as the sim-ulation progresses.

The adaptive algorithm in Fluidity-ICOM performed well over anumber of settings. Here we document some of those tried and theresults of those attempts. First, oscillations in the two prognosticGLS quantities were observed regardless of the metric used. Theultimate cause for this is that changing the mesh alters the buoy-ancy frequency due to the calculation of the vertical gradient ofdensity being approximate in a fully unstructured finite elementmesh. Adding density as a component of the metric will ensurethat more spurious changes in the density profile do not occur.


These oscillations are not significant and do not alter the results.Second, in real-world simulations density changes near the uppersurface due to heating and cooling can lead to very small areas ofunstably stratified fluid, which then causes a large spike in the ver-tical diffusivity and viscosity. This is expected model behaviourand simulates convective overturning. Although these spikes arevery short lived (around two to ten timesteps) and have little effecton the mixed layer depth, they can cause large enough spike toprevent successful solves on other prognostic fields in the model,e.g. temperature. To counter this effect it was found that increasingthe minimum dissipation of turbulent kinetic energy, �, to 10�10

prevented these spurious unstable regions giving too large valuesof diffusivity. Alone neither of these issues are large enough to cre-ate instabilities in the model, but when combined could lead toinstabilities at some adapts.

The above work presents a best estimate of suitable parametersfor the use of adaptive meshes with the GLS turbulence framework.Whilst the parameters detailed above produce excellent results,especially for the Kato–Philips study, it may be the case that otherparameters produce equally good or better results, for example alower error with fewer nodes. To fully test the parameter spaceis beyond the scope of this study, but would be a useful exerciseto carry out.

The above model simulations show that it is possible to use anadaptive algorithm in conjunction with the GLS turbulence param-eterisation framework of Umlauf and Burchard (2003). The modelperforms well when adaptivity is used with a lower average reso-lution. Moreover, the adaptive algorithm places extra nodes in theareas where extra resolution is required, namely the upper surfaceand the base of the UML. This occurs in both laboratory-scaleexperiments and in real-world ocean tests (albeit pseudo-1D).We have shown that a full 3D simulation with adaptivity and a tur-bulence parameterisation is possible, which should enable this ap-proach to be used in larger-scale ocean simulations. In addition,the ocean weather station P simulation showed that the GLSparameterisation still gives excellent agreement with observedmeasurements of MLD when the vertical resolution was 10 m. Thismakes adding the GLS parameterisation to ocean-basin scale mod-els tractable without massively increasing the number of degreesof freedom to be solved.

5. Conclusions

We have implemented and tested the GLS turbulence parame-terisation framework in a finite-element, non-hydrostatic, dynamicadaptive mesh optimising ocean model, Fluidity-ICOM. The param-eterisation performs well on both fixed and adaptive meshes. Bothlaboratory-scale and real-world simulations show excellent agree-ment with laboratory results or observed data. The adaptive algo-rithm performs as required, placing extra resolution at the surfaceand the base of the UML, whilst reducing resolution below the UMLand within the UML as appropriate This permits a vertical resolu-tion that is completely determined by the physics included in thesimulation alone, whilst reducing computational cost. A limitedsubset of the stability functions and GLS turbulence parameterisa-tions were tested with adaptivity; other combinations may changethe detail of the results presented here.

The metric to calculate the optimised mesh is flexible and threemetric pairs used in this simulation all produced reasonableapproximations to mixed layer depth behaviour in both the labora-tory and real-world tests. However, using v � q performed best atStation Papa and k�W is best in the Kato–Philips experiment. Thisunderlies the different forcing mechanisms of mixed layer depthbehaviour and underlines the importance of the physics being sim-ulated in choosing the components of the adaptivity metric.

Acknowledgements

This work was funded by NERC under Grant NE/F00270X/1 &NE/F004184/1. The authors would like to acknowledge the use ofImperial College London’s HPC service to perform the simulationspresented here. The manuscript benefited from the commentsand suggestions of anonymous reviewers.

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On the performance of a generic length scale turbulence model within an adaptive finite element ocean model