Rotating shallow water flow under location uncertainty with a structure-preserving discretization

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with a structure-preserving discretization

Rüdiger Brecht, Long Li, Werner Bauer, Etienne Mémin

To cite this version:

Rüdiger Brecht, Long Li, Werner Bauer, Etienne Mémin. Rotating shallow water flow under loca-

tion uncertainty with a structure-preserving discretization. Journal of Advances in Modeling Earth

Systems, American Geophysical Union, 2021. �hal-03131680v2�

Rotating shallow water flow under location uncertainty

1

with a structure-preserving discretization

2

R¨ udiger Brecht ¹ , Long Li ² , Werner Bauer ³ , Etienne M´ emin ²

3

1

Memorial University of Newfoundland, Department of Mathematics and Statistics,

4

St. John’s (NL) A1C 5S7, Canada

5

2

Inria/IRMAR, Campus universitaire de Beaulieu, Rennes, France

6

3

Imperial College London, Department of Mathematics,

7

180 Queens Gate, London SW7 2AZ, United Kingdom.

8

Key Points:

9

•

A physically relevant stochastic parametrization of the shallow water model is in-

10

troduced

11

•

The proposed stochastic model conserves the total energy and motivates a struc-

12

ture preserving discretization

13

•

This stochastic parametrization provides a good trade-off between model error rep-

14

resentation and ensemble spread

15

Corresponding author: R¨ udiger Brecht, [email protected]

Abstract

16

We introduce a physically relevant stochastic representation of the rotating shallow wa-

17

ter equations. The derivation relies mainly on a stochastic transport principle and on

18

a decomposition of the fluid flow into a large-scale component and a noise term that mod-

19

els the unresolved flow components. As for the classical (deterministic) system, this scheme,

20

referred to as modelling under location uncertainty (LU), conserves the global energy

21

of any realization and provides the possibility to generate an ensemble of physically rel-

22

evant random simulations with a good trade-off between the model error representation

23

and the ensemble’s spread. To maintain numerically the energy conservation feature, we

24

combine an energy (in space) preserving discretization of the underlying deterministic

25

model with approximations of the stochastic terms that are based on standard finite vol-

26

ume/difference operators. The LU derivation, built from the very same conservation prin-

27

ciples as the usual geophysical models, together with the numerical scheme proposed can

28

be directly used in existing dynamical cores of global numerical weather prediction mod-

29

els. The capabilities of the proposed framework is demonstrated for an inviscid test case

30

on the f-plane and for a barotropically unstable jet on the sphere.

31

Plain Language Summary

32

The motion of geophysical fluids on the globe needs to be modelled to get insights

33

of tomorrow’s weather. These forecasts must be precise enough while remaining com-

34

putationally affordable. Ideally they should enable to estimate likely scenarios through

35

an ensemble of physically relevant realizations, built from an accurate handling of the

36

model errors that are inescapably introduced due to physical or numerical approxima-

37

tions. To address these issues, we advocate the use of a stochastic framework to repre-

38

sent the action of the many unresolved fast/small-scale processes on the resolved flow

39

component. The derivation of the stochastic system, based on the usual conservation laws,

40

is presented in detail and simulated with an adapted structure preserving numerical model

41

to maintain numerically the nice properties of the stochastic setting inherited from a trans-

42

port principle, namely: mass and energy conservation. The versatile nature of the stochas-

43

tic derivation as well as of the proposed numerical scheme makes this framework suit-

44

able for existing dynamical cores of global numerical weather prediction models. Numer-

45

ical results illustrate the energy conservation of the numerical model and the accuracy

46

of large-scale stochastic simulations when compared to corresponding deterministic ones.

47

The ability of the random dynamical system to represent model errors is also shown.

48

1 Introduction

49

Numerical simulations of the Earth’s atmosphere and ocean play an important role

50

in developing our understanding of weather forecasting. A major focus lies in determin-

51

ing the large-scale flow correctly, which is strongly related to the parameterizations of

52

sub-grid processes (Frederiksen, O’Kane, & Zidikheri, 2013). The non-linear and non-

53

local nature of the dynamics of geophysical fluid flows make the large-scale flow struc-

54

tures interact with the smaller components. Solving the Kolmogorov scales (Pope, 2000)

55

of geophysical flows is today, and likely for a foreseeable future, completely out of reach.

56

This is due, in the first place, to the formidable computational expense that would be

57

necessary, but also to the complexity of the many fine-scale physical or bio-chemical pro-

58

cesses involved. Truncating the fine scales and simply ignoring their actions is highly detri-

59

mental to a reliable simulation of the large-scale components of the flow. Yet, an accu-

60

rate modelling of the fine-scale processes’ effects is an excruciatingly difficult task and

61

the idea of a stochastic modelling has strongly attracted the geophysical community since

62

the seminal works of (Hasselmann, 1976) and (Leith, 1975). For several years, this in-

63

terest has been strongly strengthened with the emergence of ensemble methods for prob-

64

abilistic forecasting and data assimilation issues (Berner & Coauthors, 2017; C. E. Franzke,

65

O’Kane, Berner, Williams, & Lucarini, 2015; Gottwald, Crommelin, & Franzke, 2017;

66

Majda, Franzke, & Khouider, 2008; Palmer & Williams, 2008; Slingo & Palmer, 2011).

67

The schemes proposed so far rely on very different methodological concepts. Mul-

68

tiplicative random forcing and randomization of parameters based on early turbulence

69

studies on energy backscattering (Leith, 1990; Mason & Thomson, 1992) have been pro-

70

posed (Buizza, Miller, & Palmer, 1999; Porta Mana & Zanna, 2014; Shutts, 2005). The

71

ad hoc nature of these schemes makes a systematic stochastic derivation of any flow dy-

72

namical model or configuration difficult. In addition, the absence of an explicit energy

73

balance of the noise term leads to an uncontrolled increase of variance that is potentially

74

problematic. They consequently require a proper tuning of the large-scale sub-grid model

75

and of the noise amplitude to stabilize the system. The subgrid model is, however, not

76

related to the noise term and the amplitude of the perturbations to apply is also diffi-

77

cult to specify on physical grounds. More importantly, even for low noise, an arbitrary

78

random perturbation defined outside of the physical principles on which the system has

79

been built upon may lead to strongly erroneous probability density functions of the sys-

80

tem’s dynamics (Chapron, D´erian, M´emin, & Resseguier, 2018). Other schemes based

81

on an averaging and homogenization theory have been proposed (C. Franzke, Majda, &

82

Vanden-Eijnden, 2006; C. E. Franzke & Majda, 2006) in the wake of (Majda, Timofeyev,

83

& Eijnden, 1999) and extended through the Mori-Zwanzig formalism (see the review (Gottwald

84

et al., 2017) and references therein). Those techniques are well suited for the design of

85

stochastic reduced order systems.

86

In this study, we propose to stick to a specific stochastic model, called modelling

87

under Location Uncertainty (LU) derived by (M´emin, 2014), which emerges from a de-

88

composition of the Lagrangian velocity into a smooth-in-time drift and a highly oscil-

89

lating random term. Such a slow/fast or smooth/oscillating decomposition is reminis-

90

cent to the Lagrangian decomposition introduced in the seminal work of (Andrews & McIn-

91

tyre, 1978), which is currently used for surface or internal waves studies (Kafiabad, Vanneste,

92

& Young, 2021; Salmon, 2013; Young & Jelloul, 1997; Xie & Vanneste, 2015). A sim-

93

ilar random decomposition is also at the center of the variational stochastic framework

94

of (Holm, 2015). Like our setting this latter approach applies in a broader context and

95

not only to wave solutions. Both frameworks rely on a stochastic transport principle, with

96

(Holm, 2015) dedicated to Hamiltonian dynamical systems and defined from a circula-

97

tion preserving constrained variational formulation, while (M´emin, 2014) is general and

98

built upon classical physical conservation laws.

99

This stochastic transport principle has been used as a fundamental tool to derive

100

stochastic representations of large-scale geophysical dynamics (Bauer, Chandramouli, Chapron,

101

Li, & M´emin, 2020; Bauer, Chandramouli, Li, & M´emin, 2020; Chapron et al., 2018; Resseguier,

102

M´emin, & Chapron, 2017c, 2017b, 2017a) or to define large eddy simulation models of

103

turbulent flows (Chandramouli, Memin, & Heitz, 2020; Kadri Harouna & M´emin, 2017).

104

The LU framework relies on a stochastic representation of the Reynolds transport the-

105

orem (Kadri Harouna & M´emin, 2017; M´emin, 2014) which introduces naturally mean-

106

ingful terms for turbulence studies.

107

It gathers a multiplicative random advection which is responsible for an energy backscat-

108

tering, a subgrid diffusion operator describing the mixing of the large-scale flow compo-

109

nent by the small-scale random component, and an effective advection which is attached

110

to the small scales spatial inhomogeneity. This latter term has been shown to be rem-

111

iniscent of a generalized Stokes drift component, hence designated as Itˆo-Stokes drift (Bauer,

112

Chandramouli, Chapron, et al., 2020). Backscattering and diffusion are energetically in

113

balance which leads hence to global energy conservation.

114

Recently, the LU formulation was shown to perform very well for oceanic quasi-

115

geostrophic flow models (Resseguier et al., 2017b, 2017a; Bauer, Chandramouli, Chapron,

116

et al., 2020; Bauer, Chandramouli, Li, & M´emin, 2020). It was found to be more accu-

117

rate in predicting the extreme events, in diagnosing the frontogenesis and filamentoge-

118

nesis, in structuring the large-scale flow and in reproducing long-terms statistics. Besides,

119

for a LU version of the Lorentz-63 model, derived from a Rayleigh-B´enard convection

120

in the very same way as the original model (Berge, Pomeau, & Vidal, 1987; Lorenz, 1963),

121

it has been demonstrated that the LU setting was more effective in exploring the range

122

of the strange attractor compared to classical models as well as to stochastic models built

123

with ad hoc multiplicative forcings (Chapron et al., 2018).

124

In this work, the performance of the LU representation is assessed for the numer-

125

ical simulation of the rotating shallow water (RSW) system, which can be considered as

126

the first step towards developing global random numerical weather prediction and cli-

127

mate models. In particular, this is the first time that the LU formulation is implemented

128

for the dynamics evolving on the sphere. The global energy conservation of the RSW-

129

LU system for any realization, which is analytically demonstrated here, is a strong as-

130

set of the approach and this invariant feature should be numerically conserved as closely

131

as possible. Global energy conservation is especially important for long-term climatic sim-

132

ulations. However, classical purely damping parameterizations do not take into account

133

energy and momentum fluxes from the unresolved to the resolved scales. In climatic mod-

134

els, this is believed to be a source of important biases (Gugole & Franzke, 2019).

135

Hence, we propose to combine the discrete variational integrator for RSW fluids

136

as introduced in (Bauer & Gay-Balmaz, 2019a) and (Brecht, Bauer, Bihlo, Gay-Balmaz,

137

& MacLachlan, 2019) with the numerical LU setting in order to maintain this conser-

138

vation property as well as all the transport invariants. The benefit of the proposed method

139

that relies on a modular combination of a variational integrator with a (potentially dif-

140

ferent) discretization of the LU formulation is that it should be directly applicable to ex-

141

isting dynamical cores of numerical weather prediction models.

142

The derivation of the variational integrator is based on the variational discretiza-

143

tion framework introduced by (Pavlov et al., 2011) for incompressible fluids, expanded

144

by (Gawlik, Mullen, Pavlov, Marsden, & Desbrun, 2011) to incompressible fluids with

145

advected quantities. In various papers, this framework has been further extended, for

146

instance (Desbrun, Gawlik, Gay-Balmaz, & Zeitlin, 2014) incorporated rotating and strat-

147

ified fluids of atmospheric and oceanic dynamics and (Bauer & Gay-Balmaz, 2019b) in-

148

troduced soundproof approximations of the Euler equations. Variational integrators are

149

designed by first discretizing the given Lagrangian, and then by deriving a discrete sys-

150

tem of associated Euler-Lagrange equations from the discretized Lagrangian (see (Marsden

151

& West, 2001)).

152

The advantage of this approach is that the resulting discrete system inherits sev-

153

eral important properties of the underlying continuous system, notably a discrete ver-

154

sion of Noether’s theorem that guarantees the preservation of conserved quantities as-

155

sociated to the symmetries of the discrete Lagrangian (see (Hairer, Lubich, & Wanner,

156

2006)). Variational integrators also exhibit superior long-term stability properties, cf.

157

e.g. (Leimkuhler & Reich, 2004). Therefore, they typically outperform traditional in-

158

tegrators if one is interested in long-time integration or the statistical properties of a given

159

dynamical system. Our choice for an energy preserving rather than an enstrophy con-

160

serving scheme is based on the following considerations. As shown in (Bauer, Chandramouli,

161

Li, & M´emin, 2020) for stochastic barotropic quasi-geostrophic models, using an energy

162

conserving scheme for long-term predictions yields better results than using an enstro-

163

phy conserving one. Besides, because of the direct cascade of enstrophy to high wave num-

164

bers, often stabilization through enstrophy dissipation is introduced, even in initially en-

165

strophy conserving schemes, cf. (Bonaventura & Ringler, 2005; McRae & Cotter, 2014;

166

Ringler & Randall, 2002).

167

Apart from taking into account the unresolved processes, it is paramount in prob-

168

abilistic ensemble forecasting to model the uncertainties along time (Resseguier et al.,

169

2020). In particular, operational ensemble data assimilation methods rely classically on

170

random perturbations of the initial conditions (PIC) together with an artificially care-

171

fully inflated variance (Anderson & Anderson, 1999) to increase the otherwise deficient

172

ensemble forecasts’ spread (Gottwald & Harlim, 2013; C. E. Franzke et al., 2015). Such

173

inflation has the side effect of augmenting also the representation error of the ensemble

174

members. In the present work, we compare the reliability of the ensemble spread of such

175

a PIC model with our RSW-LU system, under the same noise amplitude, and show that

176

the LU strategy yields a good trade-off between model error representation and ensem-

177

ble spread.

178

The remainder of this paper is structured as follows. Section 2 describes the ba-

179

sic principles of the derivation of the rotating shallow water system in the LU formula-

180

tion. Section 3 explains the numerical discretization of the stochastic dynamical system.

181

Section 4 discusses the numerical results for an inviscid test case with homogeneous noise

182

and a viscous test case with heterogeneous noise. In Section 5 we draw some conclusions

183

and provide an outlook for future work. In the Appendices we demonstrate the energy

184

conservation of the RSW–LU system, review some parameterizations of the noise and

185

describe the discretization of the stochastic terms.

186

2 Rotating shallow water equations under location uncertainty

187

In this section, we first review the LU representation introduced by (M´emin, 2014),

188

then we derive the rotating shallow water equations under LU, denoted as RSW–LU, fol-

189

lowing the classical strategy (Vallis, 2017). In particular, we demonstrate one important

190

characteristic of the RSW–LU, namely that it preserves the total energy of the large-

191

scale flow.

192

2.1 Location uncertainty principles

193

The LU formulation is based on a temporal-scale-separation assumption of the fol-

194

lowing stochastic flow:

195

dX t = w(X t , t) dt + σ(X t , t) dB t , (2.1)

196

where X is the Lagrangian displacement defined within the bounded domain Ω ⊂ R ^d (d =

197

2 or 3), w is the large-scale velocity that is both spatially and temporally correlated, and

198

σdB t is a highly oscillating unresolved component (also called noise) term that is only

199

correlated in space. The spatial structure of such noise is specified through a determin-

200

istic integral operator σ : (L ² (Ω)) ^d → (L ² (Ω)) ^d , acting on square integrable vector-

201

valued functions f ∈ (L ² (Ω)) ^d , with a bounded kernel ˘ σ such that

202

σ[f ](x, t) = Z

Ω

˘

σ(x, y, t)f (y) dy, ∀f ∈ (L ² (Ω)) ^d . (2.2)

203

The randomness of such a noise is driven by a functional Brownian motion B _t (Da Prato

204

& Zabczyk, 2014). The fact that the kernel is bounded, implies that the resulting ran-

205

dom flow σdB t is a centered (of null ensemble mean) Gaussian process with the well-

206

defined covariance tensor :

207

Q(x, y, t, s) = E

h σ(x, t) dB t

σ(y, s) dB s

^T

i

208

= δ(t − s) dt Z

Ω

˘

σ(x, z, t) ˘ σ

^T

(y, z, s) dz, (2.3)

209 210

where E stands for the expectation, δ is the Kronecker symbol and •

^T

denotes matrix

211

or vector transpose. The strength of the noise is measured by its variance, denoted here

212

as a, and which is given by the diagonal components of the covariance per unit of time:

213

a(x, t)dt = Q(x, x, t, t). (2.4)

214

We remark that this variance tensor has the same unit as a diffusion tensor (m ² ·s ⁻¹ )

215

and that the density of the turbulent kinetic energy (TKE) can be specified through it

216

by ¹ ₂ tr( a )/dt.

217

The previous representation (2.2) is a general way to define the noise, but other

218

formulations can be conveniently used in practice. In particular, the covariance opera-

219

tor per unit of time, Q/dt, admits an orthogonal eigenfunction basis {Φ _n (•, t)}

_n∈N

weighted

220

by the eigenvalues Λ n ≥ 0 such that P

n∈N

Λ n < ∞. Therefore, one may equivalently

221

define the noise and its variance, based on the following spectral decomposition:

222

σ(x, t) dB t = X

n ∈N

Φ _n (x, t) dβ _t ⁿ , a(x, t) = X

n ∈N

Φ _n (x, t)Φ

^T

_n (x, t), (2.5)

223

where β ⁿ denotes n independent and identically distributed (i.i.d.) one-dimensional stan-

224

dard Brownian motions. The specification of those basis functions from data driven em-

225

pirical covariance matrices enables one to construct specific noises, informed either by

226

numerical or observational data. This strategy will allow us to devise various forms of

227

the noise in the following.

228

Remark 1 Decomposition 2.1 is a temporal decomposition and not a spatial de-

229

composition as classically formulated through spatial filters and/or decimation opera-

230

tors in large-eddies simulation (LES) techniques. However, in the case of turbulent flows,

231

time and spatial scales are related. As a matter of fact, in the inertial range, the turn-

232

over time ratio for two different scales L and ` reads τ L /τ ` ∝ (L/`) ^2/3 and provides a

233

direct relation between time-scale coarsening and spatial-scale dilation. Unless specif-

234

ically needed, in the following, we will thus refer to large/small or unresolved scales with-

235

out differentiating between time or space scales. Note also that temporal filtering has

236

already been used for the definition of oceanic models (Hecht, Holm, Petersen, & Wingate,

237

2008) or large-eddies simulation approaches (Meneveau & Katz, 2000).

238

Remark 2 Decomposition 2.1 is written in terms of an Itˆo stochastic integral. This

239

decomposition could have been written in the form of a Stratonovich integral as well.

240

The calculus associated to this latter integral has the advantage of following the clas-

241

sical chain rule. However, the Stratonovich noise no longer has zero expectation. This

242

leads thus to a problematic decomposition with velocity fluctuations of non null ensem-

243

ble mean. For smooth enough integrands, it is possible to safely move from one form to

244

the other. For interested readers, more insights on the difference of the two settings and

245

their implications in stochastic oceanic modelling are provided in (Bauer, Chandramouli,

246

Chapron, et al., 2020).

247

Remark 3 The approach could be extended to express flows on arbitrary Rieman-

248

nian manifolds. In that case it is easier to work directly with the Stratonovich formu-

249

lation since it is invariant under the change of coordinates. As we consider here only flows

250

that assume the shallow approximation, the considered representation of the equations

251

in R ² and R ³ is a very accurate approximation.

252

The core of the LU model representation is based on a stochastic Reynolds trans-

253

port theorem (SRTT), introduced by (M´emin, 2014), which describes the rate of change

254

of a random scalar q transported by the stochastic flow (2.1) within a flow volume V.

255

In particular, for incompressible unresolved flows, ∇·σ = 0, the SRTT can be written

256

as

257

d t

Z

V (t)

q(x, t) dx

= Z

V (t)

D t q + q ∇· (w − w _s )

dx, (2.6a)

258

D t q = d t q + (w − w _s ) ·∇ q dt + σdB t ·∇ q − 1

2 ∇· (a∇q) dt, (2.6b)

259 260

where d t q( x , t) = q( x , t + dt) − q( x , t) stands for the forward time-increment of q at a

261

fixed point x , D t is introduced as the stochastic transport operator in (Resseguier et al.,

262

2017c) and w _s = ¹ ₂ ∇· a is referred to as the Itˆ o-Stokes drift (ISD) in (Bauer, Chan-

263

dramouli, Chapron, et al., 2020). The transport operator plays the role of the material

264

derivative in the stochastic setting. The ISD is defined by the variance tensor divergence

265

and embodies the effect of statistical inhomogeneity of the unresolved flow on the large-

266

scale component. As shown in (Bauer, Chandramouli, Chapron, et al., 2020), it can be

267

considered as a generalization of the Stokes drift associated to waves propagation with

268

the emergence of a similar vortex force and Coriolis correction. In the definition of the

269

stochastic transport operator in (2.6b), the last two terms describe, respectively, an en-

270

ergy backscattering from the unresolved scales to the large scales and an inhomogeneous

271

diffusion of the large scales driven by the variance of the unresolved flow components.

272

The diffusion term generalizes the Boussinesq eddy viscosity assumption (here with a

273

matrix eddy viscosity). This term is, nevertheless, directly related to the noise form and

274

not anymore defined by loose analogy with the molecular dissipation mechanism. The

275

backscattering term corresponds to an energy source that is exactly compensated by the

276

diffusion term (Resseguier et al., 2017c).

277

In particular, for an isochoric flow with ∇·(w − w _s ) = 0, one may immediately

278

deduce from (2.6a) the following transport equation of an extensive scalar:

279

D t q = 0, (2.7)

280

where the energy of such random scalar q is globally conserved, as shown in (Resseguier

281

et al., 2017c):

282

d t

Z

Ω

1 2 q ² dx

= 1 2 Z

Ω

q ∇· (a∇q) dx

| {z }

Energy loss by diffusion

+ 1 2 Z

Ω

(∇q)

^T

a∇q dx

| {z }

Energy intake by noise

dt = 0. (2.8)

283

Indeed, this can be interpreted as a process where the energy brought by the noise is ex-

284

actly counterbalanced by that dissipated from the diffusion term.

285

2.2 Derivation of RSW–LU

286

This section describes in detail the derivation of the RSW–LU system. This model

287

enriches the formulation described in (M´emin, 2014). Here it is fully stochastic and in-

288

cludes rotation to suit simulations of geophysical flows on a rotating frame.

289

The above SRTT (2.6a) and Newton’s second principle allow us to derive the fol-

290

lowing (three-dimensional) stochastic equations of motions in a rotating frame (Bauer,

291

Chandramouli, Chapron, et al., 2020):

292

Horizontal momentum equation :

293

D t u + f × u dt + σ

_H

dB t

= − 1

ρ ∇

_H

p dt + dp ^σ _t

+ ν∇ ² u dt + σ

_H

dB t

, (2.9a)

294

Vertical momentum equation :

295

D t w = − 1

ρ ∂ z p dt + dp ^σ _t

− g dt + ν∇ ² w dt + σ

z

dB t

, (2.9b)

296

Mass equation :

297

D t ρ = 0, (2.9c)

298

Continuity equations :

299

∇

H

· u − u _s

+ ∂ z (w − w s ) = 0, ∇

H

· σ

_H

d B _t + ∂ z σ

z

dB t = 0, (2.9d)

300301

where u = (u, v)

^T

(resp. σ

H

dB t ) and w (resp. σ

z

dB t ) are the horizontal and vertical

302

components of the three-dimensional large-scale flow w (resp. the unresolved random

303

flow σ d B _t ); f = (2 ˜ Ω sin Θ) k is the Coriolis parameter varying in latitude Θ, with the

304

Earth’s angular rotation rate ˜ Ω and the vertical unit vector k = [0, 0, 1]

^T

; ρ is the fluid

305

density; ∇

H

= [∂ x , ∂ y ]

^T

denotes the horizontal gradient; p and ˙ p ^σ _t = dp ^σ _t /dt (infor-

306

mal definition) are the time-smooth and time-uncorrelated components of the pressure

307

field, respectively; g is the Earth’s gravity value and ν is the kinematic viscosity. In the

308

following, the molecular friction term is assumed to be negligible and dropped from the

309

equations. Note that in our setting the continuity equations (2.9d) ensure volume con-

310

servation (Resseguier et al., 2017c) and mass conservation (2.9c).

311

In order to model the large-scale circulations in the atmosphere and ocean, the hy-

312

drostatic balance approximation is widely adopted (Vallis, 2017). We now specify the

313

scaling for this balance in the LU framework. We first adimensionalize the basic vari-

314

ables as

315

(x, y) = L (x ⁰ , y ⁰ ), u = U u ⁰ , t = T t ⁰ , T = L/U , z = αLz ⁰ , α = H/L, (2.10)

316

where the capital letters are used for the characteristic scales of variables and • ⁰ denotes

317

adimensional variables. To scale properly the vertical velocity, we propose to adopt a suf-

318

ficient incompressible condition (Resseguier et al., 2017c, 2017b) for the resolved com-

319

ponent in Equation (2.9d), that is

320

∇

H

· u + ∂ z w = 0, ∇

H

· u _s + ∂ z w s = 0. (2.11)

321

Note that the latter divergence-free condition on the ISD is usually considered for the

322

classical Stokes drift (J. McWilliams, Restrepo, & Lane, 2004) although being contro-

323

versial (Mellor, 2016). The three-dimensional bolus velocity introduced in the eddy-induced-

324

advection parametrization (Gent & McWilliams, 1990; Gent, Willebrand, McDougall,

325

& McWilliams, 1995; Griffies, 1998) is also assumed to be incompressible in order to pre-

326

serve the tracer’s moments. In our case, the justification of this constraint is further strengthen

327

by global energy conservation and a desirable bridge between the classical (global en-

328

ergy conserving) rotating shallow water system and its stochastic representation. Un-

329

der the condition (2.11), a classical scaling of the vertical (resolved) velocity holds:

330

w = α U w ⁰ . (2.12)

331

Apart from these classical scaling numbers, the horizontal component a

H

of the variance/diffusion

332

tensor a, which characterizes the strength of the unresolved component, is scaled as

333

a

H

= U L a ⁰

_H

, a =

a

H

a

Hz

a

Hz

a

z

, = T _σ T

EKE

MKE , (2.13)

334

where the specific factor (Resseguier et al., 2017b) is defined as the ratio between the

335

eddy kinetic energy (EKE) and the mean kinetic energy (MKE), multiplied by the ra-

336

tio between the unresolved scale correlation time T σ and the large-scale advection time.

337

From the definitions (2.3) and (2.4), the scaling of the horizontal small-scale flow reduces

338

to

339

σ

_H

d B _t = √

L ( σ

_H

d B _t ) ⁰ . (2.14)

340

In addition, we consider the following scaling between the vertical and horizontal com-

341

ponents of the unresolved flow:

342

σ

z

dB t

kσ

_H

dB t k ∼ α δ, i.e. σ

z

dB t = √

δ H (σ

z

dB t ) ⁰ , (2.15)

343

where δ is a small factor (Resseguier et al., 2017b). Again, from the definitions (2.3) and

344

(2.4), the other components of the variance/diffusion tensor scale then as:

345

a

Hz

= δ U H a ⁰

_Hz

, a

z

= δ ² α U H a ⁰

z

, i.e. a

z

ka

H

k ∼ α ² δ ² . (2.16)

346

This relation provides a ratio between the vertical and horizontal eddy diffusivities. It

347

is in practice quite small at large scale (L´evy et al., 2010, 2012).

348

Now, with f = 0 and a constant density ρ 0 , the horizontal momentum equation

349

(2.9a) implies the following scalings of the rescaled pressures:

350

˜

p = p/ρ 0 = U ² p ˜ ⁰ , d˜ p ^σ _t = dp ^σ _t /ρ 0 = √

U L (d˜ p ^σ _t ) ⁰ . (2.17)

351

Finally, substituting all the above scalings into Equation (2.9b), the adimensional ver-

352

tical momentum is given by

353

α ²

d t w ⁰ + (u ⁰ · ∇ ⁰

_H

w ⁰ + w ⁰ ∂ _z ⁰ w ⁰ ) dt ⁰ + √

(σ

H

dB t ) ⁰ · ∇ ⁰

_H

w ⁰ + δ (σ

z

dB t ) ⁰ ∂ _z ⁰ w ⁰

354

− 2

(∇ ⁰

H

· a ⁰

_H

+ δ ∂ _z ⁰ a ⁰

_Hz

) · ∇ ⁰

_H

w ⁰ + δ (∇ ⁰

H

· a ⁰

_Hz

+ δ ∂ _z ⁰ a ⁰

z

)∂ _z ⁰ w ⁰

355

+ ∇ ⁰

_H

· (a ⁰

_H

∇ ⁰

_H

w ⁰ + δ a ⁰

_Hz

∂ _z ⁰ w ⁰ ) + δ ∂ _z ⁰ (a ⁰

_Hz

∇ ⁰

_H

w ⁰ + δ a ⁰

_z

∂ ⁰ _z w ⁰ ) dt ⁰

356

= −∂ _z ⁰ p ˜ ⁰ dt ⁰ + √

(d˜ p ^σ _t ) ⁰

− dt ⁰ /Fr ² , (2.18)

357358

where Fr = U / √

gH is the Froude number. Let us now make the following assumptions:

359

α ² 1, Fr ² = O(1), = O(1), δ 1. (2.19)

360

The acceleration term on the left-hand side (LHS) of Equation (2.9b) has now a lower

361

order of magnitude than the RHS terms. Restoring the dimensions, the hydrostatic bal-

362

ance under moderate horizontal uncertainty and weak vertical uncertainty hence boils

363

down to

364

∂ z p dt + dp ^σ _t

= −ρg dt, i.e. ∂ z p = −ρg, ∂ z dp ^σ _t = 0. (2.20a)

365

We remark that the unique decomposition principle of a semimartingale process (Kunita,

366

1997) is used here to separate the bounded variation component (in terms of dt) and the

367

martingale part (in terms of dB t or dp ^σ _t ). Integrating vertically these hydrostatic bal-

Figure 1. Illustration of a single-layered shallow water system (inspired by (Vallis, 2017)). h is the thickness of a water column, η is the height of the free surface and η

b

is the height of the bottom topography. As a result, we have h = η − η

b

.

368

ances (2.20a) from 0 to z (see Figure 1), we have

369

p(x, y, z, t) = p 0 (x, y, t) − ρ 0 gz, dp ^σ _t (x, y, z, t) = dp ^σ _t (x, y, 0, t), (2.20b)

370371

where p 0 denotes the pressure at the bottom of the basin (z = 0). Following (Vallis,

372

2017), we assume that the weight of the overlying fluid is negligible, i.e. p(x, y, η, t) ≈

373

0 with η the height of the free surface, leading to p 0 = ρ 0 gη. This allows us to rewrite

374

Equation (2.20b) such that for any z ∈ [0, η] we have

375

p(x, y, z, t) = ρ 0 g η(x, y, t) − z

. (2.20c)

376

Subsequently, the pressure gradient force in the horizontal momentum equation (2.9a)

377

reads

378

− 1 ρ 0

∇

H

p dt + dp ^σ _t

= −g∇

H

η − 1 ρ 0

∇

H

dp ^σ _t , (2.20d)

379

which does not depend on z according to Equations (2.20b) and (2.20c). Therefore, the

380

acceleration terms on the LHS of Equation (2.9a) cannot depend on z, and the shallow

381

water momentum equation under weak vertical uncertainty (δ 1) can be written fi-

382

nally as

383

D

^H

t u + f × u dt + σ

_H

dB t

= −g∇

H

η dt − 1 ρ 0

∇

_H

dp ^σ _t , (2.21a)

384

D

^H

t u = d t u + (u − u s ) dt + σ

H

dB t

· ∇

H

u − 1

2 ∇

H

· a

H

∇

H

u

dt, (2.21b)

385 386

where u _s = ¹ ₂ ∇

H

·a

H

is the two-dimensional ISD and D

^H

t denotes the horizontal stochas-

387

tic transport operator whose expression is recalled in (2.21b) for the u component. The

388

relation between the unresolved flow component and the random pressure can be fur-

389

ther specified by considering a scaling of the martingale part of the momentum equa-

390

tion:

391

√ d t u ˜ ⁰ + √

(σ

H

dB t ) ⁰ · ∇ ⁰

_H

u ⁰ +

√

Ro f ⁰ × (σ

H

dB t ) ⁰ = √

∇ ⁰

_H

(dp ^σ _t ) ⁰ , (2.22)

392

where Ro = U/(f 0 L) denotes the Rossby number with f = f 0 f ⁰ , and ˜ u = u − E(u)

393

stands for the martingale part of the horizontal velocity. We note that the scaling d t u ˜ =

394

√

U d t u ˜ ⁰ is obtained from the variance of the martingale part of the vertical acceler-

395

ation term (2.18) considering the hydrostatic balance (2.20a) and the continuity equa-

396

tion (2.11). Therefore, for small Rossby number (Ro ≤ 1), the random Coriolis term

397

counter-balances the random gradient pressure force:

398

f × σ

_H

dB t ≈ − 1 ρ 0

∇

_H

dp ^σ _t . (2.23)

399

Besides, under weak vertical uncertainty, the dimensional continuity equations (2.11) and

400

(2.9d) reduce to

401

∇

H

· σ

H

dB t = ∇

H

· u _s = 0. (2.24)

402

As a result, the vertical integration (from bottom topography η b to free surface η) of the

403

continuity equations (2.9d) become

404

(w − w s )| _z=η − (w − w s )| _z=η

_b

= −h∇

H

· u, σdB t | _z=η − σdB t | _z=η

b

= 0, (2.25a)

405 406

where h = η −η b denotes the thickness of the water column (with a still bottom). On

407

the other hand, a small vertical (Eulerian) displacement at the top and bottom of the

408

fluid leads to a variation of the position of a particular fluid element (Vallis, 2017):

409

(w − w s ) dt + σdB t

_z=η = D

^H

t η, (w − w s ) dt + σdB t _z=η

b

= D

^H

t η b . (2.25b)

410 411

Combining Equations (2.25), we deduce the following stochastic mass equation:

412

D

^H

t h + h∇

H

· u dt = 0. (2.26)

413

Gathering all the elements derived so-far, we finally obtain the following RSW-LU sys-

414

tem

415

( Conservation of momentum )

416

D t u + f × u dt = −g∇η dt, (2.27a)

417

(Conservation of mass)

418

D t h + h ∇· u dt = 0, (2.27b)

419

(Random balance)

420

f × σdB t = − 1

ρ ∇dp ^σ _t , (2.27c)

421

(Incompressible constraints)

422

∇· σdB t = 0, ∇·u _s = 0, (2.27d)

423424

where the symbol H for all horizontal variables are dropped for readability reasons. In

425

Appendix A it is shown that this stochastic system conserves the global energy:

426

d t

Z

Ω

ρ

2 h|u| ² + gh ²

dx = 0. (2.28)

427

It shares thus exactly the same energy conservation property as the deterministic one

428

and beyond their formal resemblance this provides a strong physical link between the

429

two systems. Moreover, it can be noticed that under a sufficiently weak (horizontal) un-

430

certainty ( σ ≈ 0), the system (2.27) reduces to the classical RSW system, in which the

431

stochastic transport operator weighted by the unit of time, D t /dt, reduces to the ma-

432

terial derivative.

433

3 Structure-preserving discretization of RSW–LU

434

In order to perform numerical simulations of the RSW–LU (2.27) the noise term

435

σdB t has to be a priori parametrized. Its shape is conveniently expressed through a spec-

436

tral representation and a set of basis functions (2.5). In this work homogeneous as well

437

as heterogeneous spatial structures have been used and the way they are defined is re-

438

viewed in Appendix B. The incompressible homogenous noise (see Appendix B1) is de-

439

fined through a convolution kernel and is associated with Fourier modes orthogonal func-

440

tions. It is easy to implement through fast Fourier transform (FFT). As shown in Sec-

441

tion 4.1, this noise was in particular used to assess the numerical energy behavior of the

442

discrete scheme. However, homogeneous noises, although carefully scaled from a known

443

energy spectrum established at high resolution, fail to represent inhomogeneity effect en-

444

coded by spatially varying variance (the variance is constant and diagonal for homoge-

445

neous incompressible noise). This is detrimental to represent large scale effects shaped

446

by the small-scale components in geophysical fluid dynamics. As a matter of fact as shown

447

in (Bauer, Chandramouli, Chapron, et al., 2020), heterogeneous noise shapes the large-

448

scale flow in a way akin to the action of vortex force associated with the classical Stokes

449

drift.

450

In this work, two different parameterizations of heterogeneous noise have been used

451

and are described in Appendix B2. The former consists in calibrating empirical orthog-

452

onal basis functions (EOF) before the simulation (off-line) from available high-resolution

453

simulation data while the latter consists in specifying the basis functions from the on-

454

going (low resolution) simulation (i.e. on-line). The second basis functions do not de-

455

pend on data and are time evolving whereas the first ones are data driven and station-

456

ary. A procedure based on dynamic mode decomposition (Schmid, 2010) to define the

457

noise through evolving basis functions could have been as well used, as proposed by (Gugole

458

& Franzke, 2019). Such a time evolving basis, learned from a high resolution simulation,

459

are shown to perform better that stationary EOF based models. We will have the same

460

type of conclusions for the non-stationnary noise experimented here. In Section 4.2, both

461

heterogeneous noises are adopted for identifying the barotropic instability of a mid-latitude

462

jet.

463

In the following, we focus on an energy conserving (in space) approximation of the

464

random dynamical system (RSW–LU). In this context, the spatial discretization allows

465

us to mimic the balance between the global energy brought by the noise and the LU-diffusion

466

(see Eqn. 2.8) at each time step, hence no additional numerical dissipation or energy in-

467

crease is introduced into the system. Considering the definition of the stochastic trans-

468

port operator D t in (2.6b), the RSW–LU system in Eqn. (2.27a)–(2.27b) can be explic-

469

itly written as

470

d t u =

− u ·∇ u − f × u − g∇η

dt + 1

2 ∇· ∇·(au) dt − σdB t ·∇ u

, (3.1a)

471 472 473

d t h = − ∇· (uh) dt + 1

2 ∇· ∇·(ah) dt − σdB t ·∇ h

. (3.1b)

474 475

We suggest to develop an approximation of the stochastic RSW–LU model (3.1a)–(3.1b)

476

by first discretizing the deterministic model underlying this system with a structure-preserving

477

discretization method (that preserves energy in space) and, then, to approximate (with

478

a potentially different discretization method) the stochastic terms. Here, we use for the

479

former a variational discretization approach on a triangular C–grid while for the latter

480

we apply a standard finite difference method. Note that for the methodology introduced

481

in this manuscript, other spatially energy conserving discretizations rather than the sug-

482

gested variational integrator could be used too. The deterministic dynamical core of our

483

stochastic system results from simply setting σ ≈ 0 in the equations (3.1a)–(3.1b). To

484

obtain the full discretized (in space and time) scheme for this stochastic system, we wrap

485

the discrete stochastic terms around the deterministic core and combine this with an Euler–

486

Marayama time scheme.

487

Introducing discretizations of the stochastic terms that do not necessarily share the

488

same operators as the deterministic scheme has various advantages, as discussed in more

489

detail in Section 3.2.1. For instance, such a well defined interface between these two model

490

components minimizes the necessity to adapt the discretization schemes to each other

491

which, in turn, would permit us to apply our method immediately to existing dynam-

492

ical cores of global numerical weather prediction (NWP) models.

493

3.1 Discretization of deterministic RSW equations

494

As mentioned above, the deterministic model (or deterministic dynamical core) of

495

the above stochastic system results from setting σ ≈ 0, which leads via (2.4) to a ≈

496

0. Hence, Equations (3.1a)–(3.1b) reduce to the deterministic RSW equations

497

d t u =

− (∇ × u + f ) × u − ∇( 1

2 u ² ) − g∇η

dt, d t h = − ∇· (uh) dt, (3.2)

498 499

where we used the vector calculus identity u ·∇ u = (∇ ×u) × u + ¹ ₂ u ² . Note that in

500

the deterministic case d t /dt agrees (in the limit dt → 0) with the partial derivative ∂/∂t.

501

3.1.1 Variational discretizations

502

In the following we present an energy conserving (in space) approximation of these

503

equations using a variational discretization approach. While details about the deriva-

504

tion can be found in (Bauer & Gay-Balmaz, 2019a; Brecht et al., 2019), here we only give

505

the final, fully discrete scheme.

506

To do so, we start with introducing the mesh and some notation. The variational

507

discretization of (3.2) results in a scheme that corresponds to a C-grid staggering of the

508

variables on a quasi uniform triangular grid with hexagonal/pentagonal dual mesh. Let

509

T i

T i

+

T i

₋

T _j

T j

+

T _j

₋

ζ ₋

ζ ₊ e _ij

˜ e ii

₋

˜ e ii

+

˜ e _jj

₋

˜ e jj

+

Figure 2. Notation and indexing conventions for the 2D simplicial mesh.

N denote the number of triangles used to discretize the domain. As shown in Fig. 2, we

510

use the following notation: T denotes the primal triangle, ζ the dual hexagon/pentagon,

511

e ij = T i ∩ T j the primal edge and ˜ e ij = ζ + ∩ ζ ₋ the associated dual edge. Further-

512

more, we have n _ij and t _ij as the normalized normal and tangential vector relative to edge

513

e ij at its midpoint. Moreover, D i is the discrete water depth at the circumcentre of T i ,

514

η _bi the discrete bottom topography at the circumcentre of T i , and V ij = (u · n) ij the

515

normal velocity at the triangle edge midpoints in the direction from triangle T i to T j .

516

We denote D ij = ¹ ₂ (D i + D j ) as the water depth averaged to the edge midpoints.

517

The variational discretization method does not require to define explicitly approx-

518

imations of the differential operators because they directly result from the discrete vari-

519

ational principle. It turns out that on the given mesh, these operators agree with the fol-

520

lowing definitions of standard finite difference and finite volume operators:

521

(Grad n F ) ij =

⁴

F T

j

− F T

i

|˜ e ij | , (Grad t F ) ij =

⁴

F ζ

₋

− F ζ

+

|e ij | ,

(Div V ) i =

⁴

1

|T i | X

k ∈{ j,i

₋

,i

+

}

|e ik |V ik , (Curl V ) ζ =

⁴

1

|ζ|

X

˜ e

nm

∈∂ζ

|˜ e nm |V nm ,

(3.3)

522

for the normal velocity V ij and a scalar function F either sampled as F T

i

at the circum-

523

centre of the triangle T i or sampled as F ζ

_±

at the centre of the dual cell ζ _± . The oper-

524

ators Grad n and Grad t correspond to the gradient in the normal and tangential direc-

525

tion, respectively, and Div to the divergence of a vector field:

526

(∇F ) ij ≈ (Grad n F )n ij + (Grad t F )t ij , (3.4)

527

(∇ · u ) i ≈ (Div V ) i , (3.5)

528

(∇ × u) ζ ≈ (Curl V ) ζ . (3.6)

529530

The last equation defines the discrete vorticity and for later use, we also discretize the

531

potential vorticity as

532

∇ × u + f

h ≈ (Curl V ) ζ + f ζ

D ζ

, D ζ = X

˜ e

ij

∈∂ζ

|ζ ∩ T i |

|ζ| D i . (3.7)

533

3.1.2 Semi-discrete RSW scheme

534

With the above notation, the deterministic semi-discrete RSW equations read:

535

d t V ij = L

^V

_ij (V, D) ∆t, for all edges e ij , (3.8a)

536

537

d t D i = L

^D

_i (V, D) ∆t, for all cells T i , (3.8b)

538

where L

^V

_ij and L

^D

_i denote the deterministic spatial operators, and ∆t stands for the dis-

539

crete time step. The RHS of the momentum equation (3.8a) is given by

540

L

^V

_ij (V, D) =

⁴

−Adv(V, D) ij − K(V ) ij − G(D) ij , (3.9)

541

where Adv denotes the discretization of the advection term (∇ × u + f ) × u of (3.2),

542

K the approximation of the gradient of the kinetic energy ∇( ¹ ₂ u ² ) and G of the gradi-

543

ent of the height field g∇η. Explicitly, the advection term is given by

544

Adv(V, D) ij =

⁴

− 1 D ij |˜ e ij |

(Curl V ) ζ

₋

+ f ζ

₋

|ζ ₋ ∩ T i |

2|T i | D ji

₋

|e ii

₋

|V ii

₋

+ |ζ ₋ ∩ T j |

2|T j | D ij

₋

|e jj

₋

|V jj

₋

+ 1

D ij |˜ e ij |

(Curl V ) ζ

+

+ f ζ

+

|ζ + ∩ T i |

2|T i | D ji

+

|e ii

+

|V ii

+

+ |ζ + ∩ T j |

2|T j | D ij

+

|e jj

+

|V jj

+

, (3.10)

545

where f ζ

_±

is the Coriolis term evaluated at the centre of ζ _± . Moreover, the two gradi-

546

ent terms read:

547

K(V ) ij =

⁴

1

2 (Grad n F) ij , F T

i

= X

k ∈{ j,i

₋

,i

+

}

|˜ e ik | |e ik |(V ik ) ²

2|T k | , (3.11)

548

G(D) ij =

⁴

g(Grad n (D + η b )) ij . (3.12)

549550

The RHS of the continuity equation (3.8b) is given by

551

L

^D

_i (V, D) =

⁴

− Div (DV )

i , (3.13)

552

which approximates the divergence term − ∇· (uh).

553

3.1.3 Time scheme

554

For the time integrator we use a Crank-Nicolson-type scheme where we solve the

555

system of fully discretized non-linear momentum and continuity equations by a fixed-

556

point iterative method. The corresponding algorithm coincides for σ = 0 with the one

557

given in Section 3.3.

558

3.2 Spatial discretization of RSW–LU

559

The fully stochastic system has additional terms on the RHS of Equations (3.1a)

560

and (3.1b). With these terms the discrete equations read:

561

d t V ij = L

^V

_ij (V, D) ∆t + ∆G

^V

_ij , (3.14a)

562

563

d t D i = L

^D

_i (V, D) ∆t + ∆G _i

^D

, (3.14b)

564

where the stochastic LU-terms are given by

565

∆G _ij

^V

=

⁴

∆t

2 ∇ · ∇· (au)

ij − (σdB t ·∇ u) ij

· n _ij , (3.14c)

566

567

∆G _i

^D

=

⁴

∆t

2 ∇ · ∇· (aD)

i − (σdB t ·∇ D) i . (3.14d)

568

Note that the two terms within the large bracket in (3.14c) comprise two Cartesian com-

569

ponents of a vector which is then projected onto the triangle edge’s normal direction via

570

n _ij . The two terms in (3.14d) are scalar valued at the cell circumcenters i.

571

The parametrization of the noise described in Appendix B is formulated in Carte-

572

sian coordinates, because this allows using standard algorithms to calculate EOFs, for

573

instance. Likewise, we represent the stochastic LU-terms in Cartesian coordinates but

574

to connect both deterministic and stochastic terms, we will calculate the occurring dif-

575

ferentials with operators as provided by the deterministic dynamical core (see interface

576

description below). Therefore, we write the second term in (3.14c) as

577

( σ d B _t ·∇ F) ij =

2

X

l=1

( σ d B _t ) ^l _ij (∇F ) ^l _ij , (3.15)

578

in which (σdB t ) _ij denotes the discrete noise vector with two Cartesian components, con-

579

structed as described in Appendix B and evaluated at the edge midpoint ij. The scalar

580

function F is a placeholder for the Cartesian components of the velocity field u = (u ¹ , u ² ).

581

Likewise, the first term in (3.14c) can be written component-wise as

582

(∇ · ∇·(aF)) ij =

2

X

k,l=1

∂ x

k

(∂ x

l

(a kl F )) _ij

ij , (3.16)

583

where a kl denotes the matrix elements of the variance tensor which will be evaluated,

584

similarly to the discrete noise vector, at the edge midpoints. For a concrete realization

585

of the differentials on the RHS of both stochastic terms, we will use the gradient oper-

586

ator (3.4) as introduced next.

587

To calculate the terms in (3.14d) we also use the representations (3.15) and (3.16)

588

for a scalar function F = D describing the water depth. However, as our proposed pro-

589

cedure will result in terms at the edge midpoint ij, we have to average them to the cell

590

centers i.

591

In the following, we will refer to this part of the code that generates the noise on

592

a Cartesian mesh according to Appendix B as noise generation module.

593

3.2.1 Interface between dynamical core and LU terms

594

As mentioned above, the construction of the noise is done on a Cartesian mesh while

595

the discretization of the deterministic dynamical core (variational RSW scheme, Section (3.1)),

596

corresponding to a triangular C-grid staggering, predicts the values for velocity normal

597

to the triangle edges and for water depth at the triangle centers. We propose to exchange

598

information between the noise generation module (see section above) and the dynam-

599

ical core via the midpoints of the triangle edges where on such C-grid staggered discretiza-

600

tions the velocity values naturally reside. The technical details about how we realized

601

such interface in our setup are given in Appendix C.

602

This modular approach with a well defined interface between these two model com-

603

ponents has various advantages over directly implementing the noise terms on a trian-

604

gular C-grid mesh as used by the dynamical core. Firstly, this approach allows us to eas-

605

ily explore various noise types, because using a Cartesian mesh for the latter permits the

606

usage of standard algorithms for e.g. FFT or singular value decomposition (SVD). In

607

contrast, exploring these ideas directly on a triangular C-grid would significantly increase

608

the implementation work. In fact, this manuscript also serves as a proof of concept study

609

to show that such modular approach indeed works very well.

610

Moreover, the definition of an interface between the two model components should

611

minimize (or maybe even avoid) the necessity of adapting the numerics of an existing

612

deterministic core in order to incorporate the discrete stochastic LU-terms. This, in turn,

613

should allow us to apply our method directly to existing dynamical cores of NWP mod-

614

els.

615

3.2.2 Computational aspects

616

In addition to the deterministic scheme we have the terms ∆G

^V

and ∆G

^D

for the

617

RSW–LU scheme (see Eq. (3.14c) and Eq. (3.14d)). Their discretization can be differ-

618

entiated into:

619

•

The noise generation of σdB t and a. The noise generation relies on generating

620

a fixed number of pseudo-observations and carrying out a SVD to obtain the EOFs.

621

The SVD can be carried out as an economy-size SVD which depends linearly on

622

the number of triangles. Currently for LU on-line, EOFs are estimated at each time

623

step, but less frequent estimations are also possible to save computational costs.

624

•

The computation of the divergence and gradient in Cartesian coordinates. The

625

discretization of these operations are described in Appendix C, which results in

626

matrix vector multiplications.

627

Here, we obtain the discretization of ∆G

^V

and ∆G

^D

using the interface, which is

628

determined by the underlying discretization of the deterministic scheme. More specif-

629

ically, we reformulate the differential operators in Cartesian coordinates with the local

630

derivatives obtained from the deterministic scheme (see e.g. Eq. (C2)). This results only

631

in a few additional matrix vector multiplications.

632

Optimized standard methods for the noise generation on a Cartesian mesh are po-

633

tentially more efficient than a direct (and not optimized) implementation on a triangu-

634

lar mesh. Besides the advantages mentioned above and given that the additional com-

635

putational costs for interchanging the values via the interface consists of only a few ma-

636

trix vector multiplications, we advocate our modular approach rather than a direct im-

637

plementation.

638

3.3 Temporal discretization of RSW–LU

639

The iterated Crank-Nicolson method presented in (Brecht et al., 2019) is adopted

640

for the temporal discretization. Keeping the iterative solver and adding the LU terms

641

results in an Euler-Maruyama scheme, which decrease the order of convergence of the

642

deterministic iterative solver (see (Kloeden & Platen, 1992) for details).

643

To enhance readability, we denote V ^t as the array over all edges e ij of the veloc-

644

ity V ij and D ^t as the array over all cells T i of the water depth D i at time t. The gov-

645

erning algorithm reads:

646