K Brenner

R Masson

E H Quenjel

A robust VAG scheme for a two-phase flow problem in heterogeneous porous media

Keywords: phase Darcy flow, finite volume scheme, heterogeneous capillary pressure, VAG scheme MSC : 35K65, 65M08, 65M22, 6T99

A positive Vertex Approximate Gradient (VAG) scheme is proposed to discretize the total velocity formulation of two-phase Darcy flow problems in heterogeneous porous media. The discretization is based on the physical variables and allows for multiple rock types with highly contrasted petrophysical and hydrodynamical properties. The numerical experiment shows that, compared to the Phase Potential Upwind (PPU) version of VAG scheme, this new discretization is more robust and efficient in terms of nonlinear convergence

Introduction

We apply the Vertex Approximate Gradient (VAG) discretization [4] to the two-phase Darcy flow problem written in the total velocity formulation [START_REF] Brenier | Upstream differencing for multiphase flow in reservoir simulation[END_REF]. The choice of the numerical scheme is motivated by the ability of the VAG discretization to properly capture saturation jumps across different rock types. An upwind approximation [7] of the capillary diffusion term with respect to the capillary VAG fluxes is proposed in order to avoid possible undershoots and overshoots of the saturation, that could result from the non monotonicity of the VAG capillary fluxes. The time integration is chosen implicit to avoid severe time step restrictions in high velocity regions. For stability reasons, the pressure and saturation are fully coupled to account for the nonlinear transmission conditions at different rock type interfaces [START_REF] Andreianov | Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium[END_REF]. The resulting discretization can be viewed as an extension of the Hybrid Upwind (HU) transport scheme to the VAG discretization. The HU scheme was developed in the framework of the Two-Point Flux Approximation [6] in order to improve the convergence of the nonlinear solver. The authors used only one unknown per rock type interface, typically the saturation or the capillary pressure, whereas our discretization uses both pressure and saturation interface variables. This latter approach allows to accurately represent the capillary barriers. The positive approximation of the transport and capillary diffusion terms ensure a discrete maximum principle on the saturations. Following [START_REF] Brenner | Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media[END_REF], our choice of the primary unknowns at interface nodes between different rock types is based on a generalization of variable switch techniques which allows to stabilize Newton's method. As shown in the numerical section, this new HU VAG discretization provides faster nonlinear convergence than the VAG discretization based on the Phase Potential Upwinding (PPU) [START_REF] Brenner | Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media[END_REF].

Two-phase Darcy flow model

Let t f > 0 and Ω be a bounded domain of R d (d ≥ 1) such that Ω = rt∈RT Ω rt , where RT is the set of rock types. The total velocity formulation of the two-phase Darcy flow model reads

                   φ(x)∂ t s nw + div f nw V T -DΛ(x) ∇p c + (ρ w -ρ nw )g = 0, divV T = 0, V T = V nw + V w = - α∈{nw,w} η α (x, s α )Λ(x)(∇p α -ρ α g), p c = p nw -p w ∈ P c (x, s nw ), s nw + s w = 1, (2.1)
with {nw, w} denoting the set of non-wetting and wetting phases. In (2.1), φ(x) denotes the porosity, s α the phase saturation, Λ(x) the permeability tensor, p α the phase pressure, V T the total velocity, and p c the capillary pressure. The phase density ρ α is assumed constant. The gravity acceleration vector is denoted by g and its norm by g. The phase mobility function η α (x, s α ) is defined as the ratio of the relative permeability of the phase over its viscosity. Let P c (x, s) denote the monotone graph extension of the capillary pressure function (see [START_REF] Andreianov | Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium[END_REF]). It is assumed that P c and η α are spatially homogeneous in each subdomain Ω rt , rt ∈ RT ; in addition we assume that the total mobility function η(x, s) = η nw (x, s)+η w (x, 1-s) verifies η(x, s) ≥ η min > 0.We then denote by f nw the non-wetting phase fractional flow function f nw (x, s) = η nw (x, s)/η(x, s), and by D(x, s) the capillary diffusion coefficient D(x, s) = η w (x, 1 -s)f nw (x, s). The system (2.1) is completed by some initial distribution of s nw and the boundary conditions

V α • n = 0 on Γ N × (0, t f ), p α = p α Dir on Γ Dir × (0, t f ) for α ∈ {nw, w},
where n is the outward normal to Γ N , and ∂Ω = Γ N ∪ Γ Dir with Γ Dir > 0.

3 Positive VAG discretization for two-phase Darcy flows

VAG mesh, fluxes and pore volumes

The VAG discretization considers generalized polyhedral meshes of Ω [4]. Let us briefly recall some notations. Let M be the set of polyhedral cells of Ω. For each k ∈ M we denote the set of nodes of the cell k by V k and we also denote by V = k∈M V k the set of all vertices of the mesh, and by M s the subset of cells sharing the node s ∈ V. Note that the mesh is supposed to be conforming w.r.t. the partition of Ω in subdomains Ω rt , rt ∈ RT , and w.r.t. the partition {Γ N , Γ Dir } of ∂Ω. We then denote by V Dir the set of nodes located at Γ Dir .

Let

X D = {v k ∈ R, v s ∈ R, for k ∈ M
, s ∈ V} be the vector space of degrees of freedom (d.o.f.). The VAG scheme is a control volume scheme in the sense that it results, for each d.o.f. not located at the Dirichlet boundary, in a volume balance equation. The two main ingredients are therefore the conservative fluxes and the pore volumes. For

u D ∈ X D , the VAG fluxes F k,s (u D ) connect the cell k ∈ M to its nodes s ∈ V k . They can be expressed as F k,s (u D ) = s ∈V k T s,s k (u k -u s )
, with (T s,s k ) s,s ∈V k a symmetric positive definite matrix obtained from the P 1 finite element subspace defined on a tetrahedral submesh of M. As described in [START_REF] Brenner | Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media[END_REF], the portions φ k,s of each cell's pore volume k φ(x)dx is distributed to its nodes s ∈ V k \ V Dir . We then define φ k = k φ(x)dx -s∈V k \V Dir φ k,s as the remaining cell pore volume.

Choice of the primary unknowns

We recall that the mesh is conforming w.r.t. the rock type subdomains, therefore a single rock type rt k is assigned to each cell k ∈ M. We denote by χ s = {rt k , k ∈ M s } the set of rock types surrounding the node s ∈ V, and we set χ k = {rt k } for all k ∈ M.

The choice of the primary variables follows the variable switching strategy introduced in [START_REF] Brenner | Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media[END_REF]. We use the pressure of the non-wetting phase as the first primary variable for all d.o.f.; then for the cells and the nodal d.o.f. associated with a single rock type the second primary unknown is the saturation, while for the nodes s located at rock type interfaces we invoke the variable switching based on a parametrization of P c,rt , rt ∈ χ s . For such nodes we construct a set of non-decreasing continuous functions P c,χs (τ ) and S nw χs,rt (τ )) rt∈χs defined for τ ∈ [0, 1] and satisfying P c,χs (τ ) ∈ P c,rt (S nw χs,rt (τ )) for all rt ∈ χ s and τ ∈ [0, 1]; in addition we require P c,χs (τ ) + rt∈χs S nw χs,rt (τ ) to be strictly increasing. We note that this parametrization allows to deal both with a vanishing capillary diffusion and the capillary barriers. The functions τ → P c,χs (τ ), S nw χs,rt (τ )) rt∈χs have to be chosen carefully in order to improve the nonlinear solver and we refer to [START_REF] Brenner | Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media[END_REF] for a detailed discussion. In order to unify the notations, for the cells or the nodes associated with a single rock type, we introduce a trivial parametrization defined by S nw χ,rt (τ ) = τ for χ reduced to a single rock type. Given the primary unknowns p nw D = (p nw ν ) ν∈M∪V and τ D = (τ ν ) ν∈M∪V , we define

           p c,D = (p c,ν ) ν∈M∪V , with p c,ν = P c,χν (τ ν ), p w D = (p w ν ) ν∈M∪V , with p w ν = p nw ν -p c,ν , Φ α D = p α D + ρ α gZ D , with Z D = (z ν ) ν∈M∪V , s α k = S α χ k ,rt k (τ k ), k ∈ M, s α k,s = S α χs,rt k (τ s ), s ∈ V, k ∈ M s .
To sum up, as exhibited in Figure 1, the discrete phase pressure is single-valued for all d.o.f. while the saturation is single-valued for all cells (and single rock type nodes) and multi-valued at any node sharing multiple rock types.

Hybrid Upwinding (HU) VAG scheme for the diphasic model

The gravity and capillary gradient fluxes are defined by

G k,s = (ρ nw -ρ w )gF k,s Z D , C k,s = F k,s (p c,D ).
Let us introduce the total velocity fluxes V T k,s = V T k,s (p nw D , τ D ), for all k ∈ M, s ∈ V k , as well as the following discrete phase saturation functions at each d.o.f.

γ α k (τ ) = S α χ k ,rt k (τ ), k ∈ M, γ α s (τ ) = k∈Ms φ k,s φ s S α χs,rt k (τ ), s ∈ V \ V Dir .
with φ s = k∈Ms φ k,s . For N ∈ N * , we consider the time subdivision

t 0 = 0 < t 1 < • • • < t n-1 < t n • • • < t N = t f of [0, t f ].
We denote the time steps by ∆t n = t n -t n-1 for all n = 1, • • • , N . In the sequel, we omit the time superscript t n in the flux terms. Then, for a given τ 0 D ∈ [0, 1] M∪V , the scheme consists in finding (p nw,n D , τ n D ), solutions of the following system of equations for α ∈ {nw, w}:

                         φ k ∆t n γ α k (τ n k ) -γ α k (τ n-1 k ) + s∈V k f nw k,s V T k,s + D cap k,s C k,s + D g k,s G k,s = 0, k ∈ M, φ s ∆t n γ α s (τ n s ) -γ α s (τ n-1 s ) - k∈Ms f nw k,s V T k,s + D cap k,s C k,s + D g k,s G k,s = 0, s ∈ V \ V Dir , p nw,n s = p nw s,Dir , τ n s = τ s,Dir , s ∈ V Dir . (3.1)
Summing the conservation equations over the phases α ∈ {nw, w} provides the discrete divergence-free property of the total velocity fluxes

s∈V k V T k,s = 0 for all k ∈ M, k∈Ms V T k,s = 0 for all s ∈ V \ V Dir . (3.2)
Fractional flow term: We first specify two expressions of V T k,s using the upwind mobilities

V T k,s = α∈{nw,w} η α rt k (s α k )F k,s (Φ α D ) + -η α rt k (s α k,s )F k,s (Φ α D ) -, (3.3)
with x ± = max(±x, 0), or alternatively, using the cell mobilities

V T k,s = α∈{nw,w} η α rt k (s α k )F k,s (Φ α D ). (3.4)
This last choice is expected to be more stable than (3.3). Then, the fractional flow fluxes are defined by

f nw k,s V T k,s = f nw k (s nw k ) (V T k,s ) + -f nw k (s nw k,s ) (V T k,s ) -.

Capillary term:

The capillary gradient flux is not monotone. To tackle this issue, we perform a positive correction as follows

D cap k,s C k,s = η nw rt k (s nw k )η w rt k (s w k,s ) η rt k (s nw,n-1 k ) C + k,s - η nw rt k (s nw k,s )η w rt k (s w k ) η rt k (s nw,n-1 k ) C - k,s . (3.5)
Gravity term: The gravity contribution is defined similarly as in (3.5).

Remark 3.1. Note that the explicit approximation

η rt k (s nw,n-1 k
) is considered in (3.5). This choice improves the nonlinear convergence without involving any restriction on the time step.

We now state two properties of the scheme (3.1)-(3.2) that can be proved assuming that the capillary functions are bounded. We refer to [5] for the proof.

Numerical results

We test the ability of the introduced HU VAG scheme to simulate oil migration in a fractured reservoir.

It is compared to the PPU VAG version [START_REF] Brenner | Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media[END_REF] which upwinds the phase mobility w.r.t the phase potential. The domain is Ω = (0, 100 m) 3 containing a network of planar fractures of aperture 1 cm (see Figure 2). The matrix porosity is set to φ m = 0.2 and the fracture porosity to φ f = 0.4. The matrix permeability is isotropic and set to Λ m = 10 -16 m 2 while the fracture tangential permeability is set to Λ f = 10 -10 m 2 . The oil density is ρ nw = 700 Kg/m 3 and the water density is ρ w = 1000Kg/m 3 . The oil viscosity is set to µ nw = 0.005 Pa.s and the water viscosity is µ w = 0.001 Pa.s. The fracture and matrix relative permeabilities are given by k α r,f (s α ) = (s α ) 1.2 and k α r,m (s α ) = (s α ) 2 , α = nw, w. The capillary pressure is P c,m (s nw ) = -10 4 log(1-s nw ) in the matrix and P c,f (s nw ) = -10 3 log(1 -s nw ) in the fracture network. The domain is meshed using a tetrahedral mesh with 47670 cells and 1670 fracture faces. The reservoir is initially saturated with water. Dirichlet boundary conditions are imposed at the top boundary with a wetting phase pressure of 1 MPa and s w = 1, while at the lower boundary, intersected by the fracture network, we impose p w = 4 MPa and the capillary pressure resulting in the matrix saturation s nw equal to 0.9. The lateral boundaries are assumed impervious and the final time is fixed to t f = 3600 days. The time stepping is defined by ∆t 1 = ∆t init and for all n ≥ 1 by ∆t n+1 = max(∆t max , 1.2∆t n ), in case of a successful time step ∆t n , and ∆t n+1 = 0.5∆t n , if Newton's method fails to converge in 25 iterations. We have used the values ∆t init = 0.01 and ∆t max = 100 days.

We display in Figures 3456and Table 1 the results obtained for the PPU and HU VAG discretizations corresponding to the upwind mobility (3.3) labeled with HU-EtaKs and the centered one (3.4) labeled with HU-EtaK. Figure 3 shows a very good match between the PPU and HU-ETaKs schemes while Figure 4 and 5 exhibit small differences in the matrix between both HU VAG schemes.

The increased robustness of the nonlinear convergence provided by the HU VAG schemes compared with the PPU version is clearly seen in this Table 1 as well as in Figure 6. In conclusion, the positive HU VAG scheme provides similar solutions than the PPU version and exhibits an additional robustness in terms of nonlinear convergence for the simulation of highly heterogeneous media. Let us refer to [5] for more test-cases including large fracture networks.

Scheme

Figure 1 :

 1 Figure 1: Phase pressure and saturation discrete unknowns in the four cells sharing the vertex s. Each color represents a possible different rock type.

Proposition 3 . 1 .

 31 Let τ 0 D ∈ [0, 1] M∪V , then, every solution (p nw,n D , τ n D ), to the system (3.1)-(3.2) ensures τ n D ∈ [0, 1] M∪V , meaning that the saturations and the capillary pressure satisfy the physical bounds. Proposition 3.2. Let τ 0 D ∈ [0, 1] M∪V and let us assume that the total velocity fluxes are given. Then, the saturation equation (3.1) admits a solution τ n D .

Figure 2 :

 2 Figure 2: Test case configuration.

Figure 3 :

 3 Figure 3: Oil saturation volumes in the matrix (left) and in the fracture network (right) as a function of time obtained for the PPU and HU-EtaKs VAG schemes.

Figure 4 :

 4 Figure 4: Oil saturation volumes in the matrix (left) and in the fracture network (right) as a function of time obtained for the HU VAG schemes with EtaKs and EtaK.

Figure 5 :

 5 Figure 5: Oil saturation in the matrix for s nw > 0.25 and in the fractures at final time obtained for the PPU (left) and HU-EtaKs (middle) and HU-EtaK (right) VAG schemes.

Figure 6 :

 6 Figure 6: Total number of Newton iterations as a function of time for the three VAG schemes.

Table 1 :

 1 N ∆t N chop N N ewton N GM Res CPU (s) Number of the time steps N ∆t , number of the time step chops N chop , average number of Newton iterations per time step N N ewton , average number of GMRES iterations per Newton step N GM Res and CPU time for the three VAG schemes.

	PPU VAG	106	8	6.7	14.7	507
	HU-EtaKs VAG	82	0	3.9	13.4	205
	HU-EtaK VAG	82	0	4.0	13.9	203

Acknowledgment : This work was supported by the French National Research Agency under the CHARMS ANR project ANR-16-CE06-0009.


Andrew Orr

Doug Cresswell

Gareth J Marshall

Julian C R Hunt

Joël Sommeria

Chang-Gui G Wang

M Light

A "low level" explanation for the recent large warming trend over the western Antarctic Peninsula involving blocked winds and changes in zonal circulation

Keywords: Polar meteorology, 3329 Meteorology and Atmospheric Dynamics: Mesoscale meteorology, 3322 Meteorology and Atmospheric Dynamics: Land/atmosphere interactions

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

[2] Over the past 50 years the Antarctic Peninsula has experienced a warming trend of $2.8°C in near-surface air temperatures, significantly greater than the global mean warming of $0.3°C [e.g., [START_REF] Vaughan | Climate change-Devil in the detail[END_REF]. This warming is more pronounced on the western side (Figure 1). Changes to the regional oceanic circulation [START_REF] Hofmann | Water mass distribution and circulation west of the Antarctic Peninsula and including Bransfield Strait[END_REF] and the meridional component of the atmospheric circulation [START_REF] King | Climate change in the western Antarctic Peninsula since 1945: Observations and possible causes[END_REF][START_REF] Van Den Broeke | The semi-annual oscillation and antarctic climate. Part 4: A note on sea ice cover in the Amundsen and Bellingshausen Seas[END_REF] have been proposed as mechanisms for this enhanced warming trend.

[3] There has been a strengthening of the circumpolar westerly flow around Antarctica by approximately 15-20% (in the annual mean velocity) at lower levels (i.e., below the height of the Peninsula), as observed in the north of the Antarctic Peninsula by radiosonde data for the period 1969-2000 [START_REF] Marshall | Analysis of recent circulation and thermal advection in the northern Antarctic Peninsula[END_REF]. This result is consistent with the trend towards the positive polarity of the Southern Hemisphere Annular Mode (SAM) beginning in the mid-1960s [START_REF] Marshall | Trends in the Southern Annular Mode from observations and reanalyses[END_REF], which has been associated with the observed spatial pattern of Antarctic surface warming [START_REF] Kwok | Spatial patterns of variability in Antarctic surface temperature: Connections to the Southern Hemisphere Annular Mode and the Southern Oscillation[END_REF]. This index describes an annular structure in variability with synchronous pressure anomalies of opposite sign above Antarctica and the southern midlatitudes. The positive trend in the SAM indicates an increase in the strength of the circumpolar westerlies. Note that the length of this trend in the SAM is less than the period over which the warming has occured.

[4] We examine whether the interaction of the stronger circumpolar westerly flow with the Antarctic Peninsula could be partly responsible for the recent net warming experienced over its western side.

[5] [START_REF] Schwerdtfeger | Notes and correspondence: Changes of temperature field and ice conditions in the area of the Antarctic Peninsula[END_REF] noted that the advection of warm maritime air from the southeastern Pacific Ocean to the western Antarctic Peninsula had increased during the 1970's; but he did not relate this to the special features of air flow over/around the Peninsula. This flow is determined by the Froude number F defined by U/NH (where U is the velocity, H the height of the Peninsula ($2000 m), and N the atmospheric stability (typically 0.01s À1 )) and the ratio of B/L R (where B is the length of the Peninsula ($2000 km) and L R is the Rossby Deformation radius (typically 100-150 km)). When F < 0.5 the flow is blocked at lower levels by the Peninsula [O ´lafsson, 2000]. When B/L R > 1, the Coriolis acceleration deflects the flow to the right (in the Southern Hemisphere) [START_REF] Smith | Hydrostatic flow over mountains[END_REF][START_REF] O ´lafsson | The effect of rotation and surface friction on orographic drag[END_REF][START_REF] Hunt | Coriolis effects on orographic and mesoscale flows[END_REF]. If F ! 0.5 the air flow passes over the Peninsula and descends the other (easterly) side.

[6] The velocity of the circumpolar westerly flow is typically 7 ms À1 at low-level (an elevation of 850 hPa) [START_REF] Marshall | Analysis of recent circulation and thermal advection in the northern Antarctic Peninsula[END_REF], giving F % 0.35. Thus warm air below the height of the Peninsula is blocked and advected in a southerly direction along the Peninsula. The trend for stronger zonal winds implies increased blocking at lower levels, resulting in increased northerlies and enhanced warm air advection southwards, contributing to the large surface warming trend along the western Antarctic Peninsula.

[7] This concept is based on the idealised shallow-layer model of [START_REF] Hunt | Climate variability in the Amundsen and Bellingshausen Sea[END_REF] for stable mesoscale atmo-spheric flows incident on to a 'cape' which is approximate to the Antarctic Peninsula (which also discusses the limitations of numerical weather/climate (NWP) models in capturing these important mesoscale processes). In this paper we present further evidence of this theory using the results of laboratory experiments and numerical mesoscale modelling of westerly flows interacting with the Peninsula.

Laboratory Experiment

[8] The interaction of westerly zonal winds with the Antarctic Peninsula was studied using a rotating stratified tank (13 m in diameter and fluid depth of 60 cm) on the Coriolis turntable, Grenoble, France. The Antarctic Peninsula was represented by a simple 'cape' consisting of a radial barrier (perpendicular to the mean flow) of length perpendicular to the wind B = 100 cm, height H = 10 cm and width parallel to the flow D = 35-50 cm, abutting onto a bounding wall (parallel to the mean flow) of height 20 cm. The cape was placed on the tank bottom. Westerly and easterly flows over the cape of 1 -10 cms À1 were driven by either impulsively increasing or decreasing the turntable rotation rate, so that the Froude number F varied from 0 to 2.

[9] Figure 2 shows horizontal streamline patterns of lowlevel westerly flow relative to the cape. Here F % 1/3 and L R = 45 cm (i.e., B/L R > 1). The upstream flow divides in the horizontal plane when blocked by the cape, with a stagnation point in the flow at y = y S (y = 0 at the bounding wall and y = B at the tip of the cape). For 0 < y < y S the streamlines turn to the right along the cape and recirculate in the upwind direction. For y > y S the streamlines turn to the left, flowing around the cape and forming a jet extending downstream and a circulation cell in the lee of the cape. The proportion of the flow recirculating upwind was y S /B % 0.6. This appeared to depend on the mean upwind flow speed, with more circulating at higher Froude number (as expected theoretically [START_REF] Hunt | Coriolis effects on orographic and mesoscale flows[END_REF]). Because of the low-Froude number, the upper-level flow (i.e., above the height of the cape) passed over the cape with little deflection (not shown). Further experimental runs showed that in all high-Froude number flow (F ! 0.5) the streamlines pass over the cape and descended on the lee side (not shown).

Numerical Mesoscale Modelling

[10] The numerical model used in this study is the atmosphere-only UK Met. Office Unified Model, version 4.5 (UM 4.5). The UM is a General Circulation Model (GCM) system designed to run as a climate, global, or limited-area mesoscale model. UM 4.5 implements a nonlinear, hydrostatic set of dynamical equations on a horizontal latitude-longitude grid and a 'hybrid'sigma/pressure coordinate system in the vertical [START_REF] Cullen | The unified forecast/climate model[END_REF]. Details of the physical parameterizations can be found in [START_REF] Clark | One-dimensional site-specific forecasting of radiation fog. Part 1: Model formulation and idealised sensitivity studies[END_REF]. For the simulations a nested limited-area domain with horizontal resolution of 12 km and 38 vertical levels was centred over the Antarctic Peninsula. Use of a rotated coordinate pole achieves uniform resolution. Lateral boundary conditions are generated initially from UM 4.5 running as a global model. [11] Figure 3a shows the 10m-wind speed and surface pressure (hPa) over the Antarctic Peninsula for 29 January 2002 06UTC computed using UM 4.5. The upwind flow is well defined and representative of the westerly flow around Antarctica (F % 0.3). The computations show upstream flow divergence in the horizontal plane with a stagnation point in the flow at y = y S (y = 0 at the base of the Peninsula and y = B at the tip of the Peninsula). Here y S /B % 0.6. For 0 < y < y S the flow moves south along the Peninsula and recirculates in the upwind direction. For y > y S the flow moves north. The beginnings of a jet forming downstream of the Peninsula is rapidly deflected to the north by a low pressure system over the Weddell Sea. The upper level flow passes over the Peninsula with little deflection (not shown).

[12] The general flow field and the location of the stagnation point corresponds closely to the laboratory experiment discussed earlier. Figure 3b shows the surface temperature at 10m, illustrating that relatively warm northerly air (2 -4°C) at low-level is being transported to the Peninsula. As the mean upwind flow speed increases, i.e., as the Froude number increases towards 0.5, it was found that the stagnation point y S moves north towards the tip of the Peninsula and the size of the recirculating region (y S /B) increases. When the Froude number exceeds 0.5 the deflection of the flow is reduced and the flow passes over the Peninsula and descends on the downwind side (not shown).

Discussion

[13] The laboratory experiment and numerical mesoscale simulations show that low-level, low-Froude number (F < 0.5), westerly flow is blocked by the cape/Antarctic Peninsula. As B/L R > 1 the flow divides in the horizontal plane with some deflected to the right by the Coriolis acceleration and recirculating upwind, and some flowing around the cape and forming a jet extending downstream. The simulations demonstrate that the proportion of the flow recirculating upwind depends on the mean upstream flow speed, with more circulating at higher upstream speeds, and that upper-level, low-Froude number flow is relatively unde-flected by the obstacle. This blocking mechanism is only partially evident from numerical modelling using a coarse horizontal resolution (i.e., NCEP/NCAR reanalysis data, data from UM 4.5 running as a climate model [START_REF] Connolley | The Antarctic climate of the UKMO unified model[END_REF]).

[14] This local flow structure interacts with the increased and more frequent circumpolar westerly flow around Antarctica to contribute to the enhanced near-surface warming over the western Antarctic Peninsula. The low-level blocking of the flow here gives rise to northerly winds and greater transport of relatively warm air along the western side (and to a limited extent to the eastern side of the Peninsula) (see Figures 2 and3). As the velocity of the westerly winds strengthens, since the scale of the recirculating flow region y S /B increases the airstream moving along the western side of the Peninsula originates from a more northerly latitude in the Pacific Ocean. Hence the advected air is increasingly warmer. This hypothesis is supported by the observed increase in precipitation events on the western coast [START_REF] Turner | Variability of precipitation over the coastal Antarctic Peninsula from synoptic observations[END_REF]. Another observation consistent with the proposed 'low-level' explanation for the Peninsula warming is that [START_REF] Marshall | Changes in Antarctic Peninsula tropospheric temperatures from 1956 to 1999: A synthesis of observations and reanalysis data[END_REF] showed that the mid-tropospheric warming above the north-western Peninsula was only approximately half that recorded at the surface.

[15] During the past few decades the Peninsula surface temperature record shows a warming that increases with southerly latitude [START_REF] Vaughan | Climate change-Devil in the detail[END_REF][START_REF] Shuman | Decadal-length composite inland west Antarctic temperature records[END_REF]. The trend varies from 0.031°C yr À1 at 62°S to 0.110°C yr À1 at 76°S, as average temperature decreases (cf. Figure 1), consistent with warm air being advected south to progressively colder regions. Associated with the warming trend along the west coast is a decrease in the sea-ice extent [Jacobs and Comiso, 1997], which has the effect of amplifying climatic fluctuations through the ice-albedo feedback mechanism. The proposed changes in warm air advection and the resultant increase in northerlies would reduce sea-ice extent by increasing melting and suppressing the northward expansion of the sea-ice, respectively. [16] The blocking-mechanism could explain why [START_REF] King | Recent climate variability in the vicinity of the Antarctic Peninsula[END_REF] observation that annual near-surface temperatures along the western peninsula are more highly correlated with meridional rather than zonal flow over the 1945 -90 period. Coincident trends in tropospheric winds above Bellingshausen in the northern Peninsula (where the reduced orography means blocking is less significant) show increases in zonal but not meridional flow [START_REF] Marshall | Analysis of recent circulation and thermal advection in the northern Antarctic Peninsula[END_REF]. This dataset also reveals a greater increase in the westerlies during summer than winter during 1969 -2000, consistent with the observed changes in the SAM.

[17] In reality, the 'climatological' flow conditions simulated in the laboratory are relatively rare: typically the flow around Antarctica also includes synoptic scale eddies and fronts. Additionally, the laboratory simulations will not capture the complicated interaction between katabatic winds blowing off the Peninsula and the West Antarctic Ice Sheet and the prevailing westerlies, as evident in Figure 3a. Nonetheless, typically, there are north-westerlies affecting the western side of the Peninsula as the Bellingshausen Sea often is a region of strong cyclolysis. This is also consistent with the effects of the topographic barrier, as is the observation of a mean northerly meridional wind component at an elevation of 850 hPa for 11 months of the year, derived from a limited radiosonde dataset from the western coast (Rothera) during 2002 -03.

[18] The increase in the strength of the westerlies also makes the probability of high-Froude number flow (F ! 0.5) more likely. The laboratory and numerical simulations showed that all high-Froude number flow passes over the Peninsula and descends on the lee (easterly) side [START_REF] Hunt | Coriolis effects on orographic and mesoscale flows[END_REF][START_REF] Hunt | Climate variability in the Amundsen and Bellingshausen Sea[END_REF], transporting relatively warm air to the eastern side of the Antarctic Peninsula. However, it could be expected that the importance of this mechanism for influencing eastern Peninsula climate will remain comparatively weak [START_REF] Schwerdtfeger | Notes and correspondence: Changes of temperature field and ice conditions in the area of the Antarctic Peninsula[END_REF].

[19] Finally, this mechanism may also explain other anomalies in local climate variation in other areas of the world with pronounced orography and where there are trends in the large scale winds.

[20] Acknowledgments. This work was supported by grants from NERC to the Centre for Polar Observation & Modelling, University College London. The laboratory work was supported by the EEC contract 'Transnational Access to Major Research Infrastructures' HPRI CT 2001-00168.

Figure 1 .

 1 Figure 1. A map of the Antarctic Peninsula region showing temperature information at research stations mentioned in the text. The annual temperature trend is given in °C yr À1 , and the statistical significance, if any, given in parantheses. The mean annual temperature and length of the observation record are given in italics.

Figure 2 .

 2 Figure 2. Horizontal streamline photograph of simulated low-level westerly flow (left to right) around a cape in the Southern Hemisphere, with F % 1/3. The length y S defines the position of the stagnation point from the base of the cape and B is the length of the cape. See color version of this figure in the HTML.

Figure 3 .

 3 Figure 3. Atmospheric westerly flow over the Antarctic Peninsula at 29 January 2002 06UTC. Computed using UM 4.5 with a horizontal resolution of 12 km. Shown are (a) 10 m-wind speed (ms À1 ) vectors (every 8 grid points) and mean surface level pressure (hPa) and (b) surface temperature (°C). The length y S defines the position of the stagnation point from the base of the Peninsula and B is the length of the Peninsula. See color version of this figure in the HTML.

Department of Space and Climate Physics, University College London, London, UK.

British Antarctic Survey, Cambridge, UK.

Laboratoire des Ecoulements Ge ´ophysiques et Industriel/CORIOLIS, CNRS, Grenoble, France.

Department of Meteorology, University of Reading, Reading, UK.