Florian Bernard

Angelo Iollo

Gabriella Puppo

A Local Velocity Grid Approach for BGK Equation

Keywords: Boltzmann equation, BGK model, DVM Equation de Boltzmann, modèle BGK, DVM

Solving complex flows with BGK equation and the Discrete Velocity Method (DVM), may require a large number of velocity grid points and hence the resolution becomes very slow. However, locally, in each space cell, the distribution function is supported only on a sub-set of the velocity grid and is zero elsewhere. Thus, solving the equation for such velocities is useless. A local adaptive velocity grid approach is proposed and compared to a "global velocity grid" BGK solution. Simulations in 1D and 2D are presented for different Knudsen numbers showing the computational gain of the proposed approach.

Introduction

In hydrodynamic regimes, fluid flows can be simulated thanks to standard models such as Navier-Stokes or compressible Euler equations. However, some regimes cannot be qualified as hydrodynamic and the continuum equations are not able to correctly describe the dynamics of the flow. The parameter that dictates whether or not a flow is hydrodynamic is the Knudsen number Kn. It is defined as the ratio between the mean free path λ between the particles and the characteristic length of the physical problem L. When this number goes towards zero, the hydrodynamic regime is reached. For large Knudsen number (usually higher than 10 -2 ), the regime is qualified as rarefied and the governing equation is the Boltzmann equation [START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF].

Simulating rarefied gas flows for realistic test cases is a challenging problem since the collision term of the Boltzmann equation is very costly to compute. In particular, solving directly the Boltzmann equation is usually computationally prohibitive because of the complexity of the collision operator. Several techniques exist to reduce the cost of this term like Direct Simulation Monte-Carlo [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF], which is a particle solver especially efficient in the rarefied regime. However, the computational time requirement increases very rapidly [START_REF] Pareschi | An introduction to the numerical analysis of the Boltzmann equation[END_REF] as the hydrodynamic regime is reached. The number of collisions increases and a strong restriction appears on the time step. For this reason, attempts have been made to derive numerical solvers for the Boltzmann equation which are not based on particles [START_REF] Tcheremissine | Conservative evaluation of Boltzmann collision integral in discrete ordinates approximation[END_REF]. A particularly successful model is the BGK model [START_REF] Bhatnagar | A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems[END_REF], in which the collision term of Boltzmann equation is greatly simplified. It models the collision term as a relaxation towards a Maxwellian distribution function. More recently, an extension to BGK has been widely studied to include thermal effects which are not correctly represented in the standard BGK model. It is the ES-BGK model [START_REF] Andries | The Gaussian-BGK model of Boltzmann equation with small Prandtl number[END_REF].The ability of this model to correctly approximate the full Boltzmann equation had been studied in [START_REF] Andries | Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases[END_REF]. For small relaxation time, a strong restriction on the time step exists also for the BGK model, but it can be easily treated by using implicit schemes, such as IMEX ( [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF], [START_REF] Kennedy | Additive Runge-Kutta schemes for convectiondiffusion-reaction equations[END_REF], [START_REF] Coron | Numerical passage from kinetic to fluid equations[END_REF]). Recently, this technique has been extended to the efficient integration of the ES-BGK model ( [START_REF] Alaia | A hybrid method for hydrodynamic-kinetic flow -Part II -Coupling of hydrodynamic and kinetic models[END_REF], [START_REF] Filbet | An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation[END_REF]). IMEX numerical schemes provide efficient solvers which give the correct asymptotic properties ( [START_REF] Jin | Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations[END_REF], [START_REF] Filbet | A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources[END_REF]). Such models are used usually with the discrete velocity method (DVM) [START_REF] Buet | A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics[END_REF], that requires a discretization of the velocity space. The same global velocity grid is employed for each space discretization point. A major bottleneck of such an approach is the size of the velocity grid. Indeed, the boundaries of the velocity grid have to be determined according to the maximum velocity and temperature in the flow while its spacing depends on the minimum temperature. In realistic test cases like hypersonic atmosphere re-entries or satellite engines, very strong gradients in temperature can be observed. The solution needs a very large and fine grid. Thus the computational requirements increase quickly and make the phenomenon difficult to simulate especially in 3D.

This important problem has already been addressed in [START_REF] Chen | A unified gas kinetic scheme with moving mesh and velocity space adaptation[END_REF]. There, an AMR technique is presented in velocity space for the unified gas-kinetic scheme [START_REF] Xu | A unified gas-kinetic scheme for continuum and rarefied flows[END_REF]. A careful work has been carried out on the error estimation due to quad/octree meshes to make the method accurate. Since the grids are not connected, interpolation is required in the transport step, to match the velocity grids in the neighbouring space cells. This interpolation can lead to approximation errors and possible increasing computational time. In [START_REF] Baranger | AIP Conference Proceedings[END_REF] a criteria to decide whether or note the grid has to be refined or coarsened is presented and proves to be efficient with and AMR algorithm for steady flows. However, dealing with unsteady flows requires a grid adaptation at each time step and thus, interpolation of the solution between two successive grids.

In this paper, a local velocity grid approach is presented for the BGK model. The idea is to build thanks to simple algorithms local velocity grids in each space cell, keeping a high accuracy and saving as much computational time as possible. The proposed approach is at worse as expensive as a standard DVM.

The new approach presented here avoids interpolations and refinement steps. Our numerical tests show that this method significantly decreases the computational time since the velocity grid is often drastically reduced with a negligible overhead. The efficiency of the method will be presented on 1D and 2D test cases for different regimes.

Governing equations

In this section, we detail the full model and its 1D and 2D reductions.

The BGK model

A typical property of the Boltzmann equation is given by the Chapman-Enskog expansion [START_REF] Chapman | The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases[END_REF]. It ensures the asymptotic limit towards Navier-Stokes equation for small Knudsen numbers (Kn ≃ 10 -3 ) and towards compressible Euler equations for Kn → 0. The BGK model respect this property and is an approximation of the Boltzmann equation particularly viable for moderate Knudsen number, in the kinetic regime (roughly Kn ≤ 1). This model gives a simple representation of the collision term:

∂f ∂t (x, ξ, t) + ξ • ▽ x f (x, ξ, t) = 1 τ (M f (x, ξ, t) -f (x, ξ, t)) (1)
where τ is the relaxation time and f is a density distribution function that depends on space x ∈ R d , microscopic velocity ξ ∈ R N , and time t with f 0 (x, ξ, t = 0) as initial data. M f is a Maxwellian distribution function obtained as follows:

M f (x, ξ, t) = ρ(x, t) (2πRT (x, t)) N/2 exp - | ξ -U (x, t) | 2 2RT (x, t) (2)
where R is the universal gas constant and T (x, t), U (x, t) and ρ(x, t) are macroscopic values of temperature, velocity and density calculated from the moments of f defined by:

  ρ(x, t) ρ(x, t)U (x, t) E(x, t)   = R N f (x, ξ, t)m(ξ)dξ with m(ξ) =    1 ξ 1 2 | ξ | 2    (3)
E(x, t) is the total energy obtained as follows:

E(x, t) = N 2 ρRT (x, t) + 1 2 ρ(x, t) | U (x, t) | 2 (4)
We consider a monoatomic gas for which the ratio of specific heats γ can be calculated as:

γ = 1 + 2 N
In the following, N = 3 and therefore γ=5/3.

The relaxation time for the BGK model can be written as:

τ -1 = cρT 1-ν

Inria

where c is a constant that depends on the reference conditions and ν is the exponent of the viscosity law of the gas. The constant c can be expressed as :

c = RT ν 0 µ 0
where µ 0 is the reference viscosity of the gas at the reference temperature T 0 .

In the following, we will consider the BGK equation in dimensionless form, that is:

∂f ∂t (x, ξ, t) + ξ • ▽ x f (x, ξ, t) = 1 τ (M f (x, ξ, t) -f (x, ξ, t)) (5)
M f (x, ξ, t) = ρ(x, t) (2πT (x, t)) N/2 exp - | ξ -U (x, t) | 2 2T (x, t) (6)
τ -1 = Kn ∞ ρT 1-ν (7)
ρ, U and T are now the dimensionless macroscopic quantities and Kn ∞ is the Knudsen number in reference conditions.

For 1D and 2D cases, the non leading dimension can be dropped in velocity (two in 1D and one in 2D) by integrating the distribution function directly along these dimensions. Two distribution functions φ and ψ of lower dimensions are obtained to create the reduced model (see [START_REF] Chu | Kinetic-theoretic description of the formation of a shock wave[END_REF] and [START_REF] Bernard | Second-order Euler-preserving wall condition for the BGK equation on Cartesian meshes[END_REF] for more details). For instance in 1D with ξ = (ξ u , ξ v , ξ w ):

     ∂ t φ(x, ξ u , t) + ξ u ∂ x φ(x, ξ u , t) = 1 τ (M φ (x, ξ u , t) -φ(x, ξ u , t)) ∂ t ψ(x, ξ u , t) + ξ u ∂ x ψ(x, ξ u , t) = 1 τ (M ψ (x, ξ u , t) -ψ(x, ξ u , t)) (8)
where

φ = R 2 f dξ v dξ w , ψ = R 2 1 2 (ξ 2 v + ξ 2 w )f dξ v dξ w , M φ = R 2 M f dξ v dξ w and M ψ = R 2 1 2 (ξ 2 v + ξ 2 w )M f dξ v dξ w .

The discrete model in velocity space

By construction the distribution function f and the corresponding Maxwellian distribution function M f satisfy:

R 3 M f m(ξ)dξ =   ρ(x, t) ρ(x, t)U (x, t) E(x, t)   = R 3 f m(ξ)dξ
This is essential to ensure conservation of mass, momentum and energy because multiplying (1) by m(ξ) and integrating in velocity space one recovers the conservation laws for mass, momentum and energy:

∂ t R 3 m(ξ)f (ξ)dξ + ∂ x R 3 ξm(ξ)f (ξ)dξ = 0
In the discrete case, a grid must be introduced in velocity space and the integrals are evaluated by quadrature. Let ., . denote the quadrature rule and:

  ρ ρU E   = f (ξ), m(ξ)
be the discrete moments of f . Now:

M f (ξ), m(ξ) =   ρ ρU E   = f (ξ), m(ξ)
due to the error introduced by the quadrature rule and conservation would not hold exactly.

In [START_REF] Mieussens | Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries[END_REF], Mieussens proved that a discrete Maxwellian can be expressed as M f = exp(α • m(ξ)) and such that:

M f (ξ), m(ξ) =   ρ ρU E   (9)
This form holds true also in the continuous case for the exact Maxwellian with:

α c = ln( ρ(x, t) (2πRT (x, t)) d/2 ) - |U (x, t)| 2 2RT (x, t) , U (x, t) RT (x, t) , - 1 RT (x, t)
For the 1D reduced model, this expression becomes :

M φ , m 1 (ξ) + M ψ , e 3 = f, m 1 (ξ) (10)
with e 3 = (0, 0, 1) T and m

1 = (1, ξ u , 1 2 |ξ U | 2 ) T .
In 2D, the discrete Maxwellian of φ and ψ is such as:

M φ , m 2 (ξ) + M ψ , e 4 = f, m(ξ) (11)
with e 4 = (0, 0, 0, 1) T and m 2 = (1, ξ u , ξ v ,

1 2 |ξ U + ξ v | 2 ) T .
Equation [START_REF] Chapman | The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases[END_REF] and [START_REF] Chen | A unified gas kinetic scheme with moving mesh and velocity space adaptation[END_REF] are a non linear system in α which can be solved with a Newton-Raphson method. The solution in the continuous case (α c ) will be the starting value for the Newton-Raphson algorithm.

In our case, the trapezoidal quadrature rule is used because it has spectral accuracy for smooth and periodic functions on a uniform grid. For this reason we use a uniform grid symmetric with respect to zero and such that f is negligible outside the grid. In 1D, the velocity grid is defined as:

G = (ξ i ) i=-nv ..nv with ξ i = i∆ξ
For multidimensional cases, the same discretization is performed independently in all directions.

The numerical method

Since the two equations for φ and ψ have the same structure, the numerical method will be presented for one equation only where the function f will stand for φ and ψ and M f the corresponding Maxwellian. We use a finite volume method and it will be described for the 2D reduced model.

Inria

The space discretization

The space Ω is discretized on a Cartesian grid with n × m cells:

Ω = i=1..n j=1..m Ω i,j = i=1..n j=1..m [x i-1/2 , x i+1/2 ] × [y j-1/2 , y j+1/2 ]
such that (x i , y j ) are the coordinates of the center of the cell (i, j) and (x i+1/2 , y j ) are the coordinates of the center of the interface between cells (i, j) and (i + 1, j). Eq.( 5) is integrated

on the space cell Ω i,j = x i - ∆x 2 , x i + ∆x 2 × y j - ∆y 2 , y j + ∆y 2 : ∂f i,j ∂t + ξ • ∂Ωi,j f n ∂Ωi,j dσ = 1 τ i,j (M fi,j -f i,j ) (12)
where f i,j = 1 |Ω i,j | Ωi,j f dxdy and M fi,j = M f (x i , y j ) the discrete Maxwellian compute from [START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF].

Since a uniform Cartesian grid is considered, the equation can be simply rewritten in terms of fluxes at each numerical interface:

∂f i,j ∂t + 1 ∆x (F i+ 1 2 ,j -F i-1 2 ,j + F i,j+ 1 2 -F i,j-1 2 ) = 1 τ i,j (M fi,j -f i,j ) (13)
with F i+ 1 2 ,j the numerical flux between cell Ω i,j and cell Ω i+1,j (with a similar notation for the other fluxes) which is expressed as :

F i+ 1 2 ,j = max(0, ξ)f l + min(0, ξ)f r (14)
with f r and f l the values of f on the two sides of the interface. The numerical expression of the distribution functions f l , f r depends on the reconstruction used at the numerical interface. For a first order reconstruction, f l = f i,j and f r = f i+1,j . One can also use a MUSCL reconstruction with slope limiters (MinMod for example) to get second order accuracy:

     f l = f i,j + 1 2 MinMod(f i+1,j -f i,j , f i,j -f i-1,j ) f r = f i+1,j - 1 2 MinMod(f i+1,j -f i,j , f i+2,j -f i+1,j ) (15)

The time discretization

The time discretization can be performed explicitly for all terms. But in this case, the time step would be determined by the space discretization (∆x), the maximum velocity of the velocity grid and the relaxation time τ . For small Knudsen numbers, the relaxation part becomes very stiff (τ very small) and imposes a very strong restriction on the time step. For this reason, the IMEX scheme [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF], [START_REF] Pareschi | Implicit-Explicit Runge-Kutta Schemes and Applications to Hyperbolic Systems with Relaxation[END_REF] is chosen. Here the relaxation term is treated implicitly while the convective part is non stiff an explicit scheme can be more efficient.

The time integration for a ν-stages IMEX Runge-Kutta scheme [START_REF] Kennedy | Additive Runge-Kutta schemes for convectiondiffusion-reaction equations[END_REF] applied to (13) reads as follows:

f n+1 i,j = f n i,j -∆t ν k=1 ωk ξ∇ x f (k) i,j + ∆t τ i,j ν k=1 ω k (M (k) fi,j -f (k) i,j ) f (k) i,j = f n i,j -∆t k-1 l=1 Ãk,l ξ∇ x f (l) i,j + ∆t τ i,j k l=1 A k,l (M (l) fi,j -f (l) i,j ) f (1) i,j = f n i,j + ∆t τ i,j A 1,1 (M (1) fi,j -f (1) i,j ) (16)
where A and à are ν × ν matrices, with Ãi,s = 0 if s ≥ i and A i,s = 0 if s > i. These coefficients are derived from a double Butcher's tableaux:

à ωT A ω T
It is worth to remark that to compute the stage k (f

(k) i,j
), we need to know the Maxwellian at stage k which depends a priori on the moments of f (k) i,j . But we observe that if we compute the moments of f

(k) i,j : < m(ξ), f (k) i,j > =< m(ξ), f n i,j > -∆t k-1 l=1 Ãk,l < m(ξ), ξ∇ x f (l) i,j > + ∆t τ k l=1 A k,l < m(ξ), (M (l) fi,j -f (l) i,j ) > =< m(ξ), f n i,j > -∆t k-1 l=1 Ãk,l < m(ξ), ξ∇ x f (l) i,j > because the moments of M (l)
fi,j and f (l) i,j are the same. The moments of f

(k)
i,j can be then computed using already known quantities and therefore the discrete Maxwellian M (k) fi,j at stage k is obtained. All the quantities until stage k -1 are known so the equation for stage k becomes:

f (k) i,j = τ i,j A k,k ∆t + τ i,j f n i,j -∆t k-1 l=1 Ãk,l ξ∇ x f (l) i,j + ∆t τ i,j k-1 l=1 A k,l (M (l) fi,j -f (l) i,j )+ A k,k ∆t τ i,j M (k) fi,j f (k)
i,j can be then computed explicitly since the right hand side is known.

In our case, we will be interested in first and second order scheme.

IMEX 1st order scheme 0 1 1 1

Explicit scheme Implicit scheme

Inria IMEX 2nd order scheme

0 0 0 0 0 0 0 1 0 0 1 2 1 2 1 2 0 0 -1 2 1 2 0 0 1 2 1 2 0 1 2 1 2

Explicit scheme

Implicit scheme

A local velocity grid approach

Here, the local velocity approach is presented. In a first part, the algorithm will be described in 1D. Then the approach will be extended to a multidimensional space.

The 1D algorithm

We start describing the construction of the local velocity grid in 1D. For test cases with strong shocks or with high temperature or velocity differences, the boundaries of the velocity grid have to be very extended with a fine grid spacing. Usually the grid is defined with the following conditions:

G = (ξ i ) i=-nv ..nv such that    ξ max = |U | max + 5 √ T max ξ min = -ξ max ∆ξ ≤ √ T min (17)
with |U | max the maximum absolute value of the macroscopic velocity and T max the maximum temperature over space and time. The velocity grid is then taken symmetric with respect to zero. The first condition ensures that most of the distribution function (more than 99.99%) is contained in the grid for every space cell at any time t if it is not too far from the Maxwellian distribution function. The last condition ensures that the Maxwellian distribution function will be always defined on at least three grid points. In spatial cells, the distribution function will be concentrated on a sub-part of the global velocity grid and will be zero (or negligible) elsewhere. The idea of the present work is to avoid the computation of the distribution function where it is negligible and more specifically, where the convective fluxes will be zero or negligible.

Local grids in velocity are chosen for each space cell based on the macroscopic quantities. Each local grid is a sub-set of the global velocity grid which would be the one chosen thanks to [START_REF] Kennedy | Additive Runge-Kutta schemes for convectiondiffusion-reaction equations[END_REF].

In particular, considering a global velocity grid with a set of N v velocity points (ξ k ) k=1,Nv , the local grid on a cell i is set as follows:

ξi min = ξ l such that ξ l = max k∈[1,Nv] (ξ k ≤ U i -α T i ) ξi max = ξ l such that ξ l = min k∈[1,Nv] (ξ k ≥ |U i | + α T i ) (18)
where U i and T i are the macroscopic velocity and temperature in cell i. ξi min and ξi max are respectively the minimum and maximum velocity node contained in the local grid for the space cell i. Note that in this way, the interfaces of velocity cells belonging to different local grids are aligned. Then, no interpolation is needed to compute the numerical fluxes in the transport step. Indeed, all the local grids are based on the global grid. Thus, ξ is outside the local grid at t n , f (x, ξ, t n ) is set to zero.

Defining in such a way the local velocity grids is not enough as the gas evolves. Let us consider the case where a velocity point ξ k exists in the velocity grid of a given space cell but not in the velocity grid of the neighbour cell. Even small, the value of the distribution function f i,k is not zero in the first cell while f i+1,k = 0. Thus a small flux is created and it must contribute to update f i,k and f i+1,k . But for the cell that does not contain this microscopic velocity, the value of the distribution function cannot be updated unless that point is included in the local grid and may become non negligible in long time. Hence, at the beginning of the time step, the local grids will be set at the union of the local grids contained in the stencil of the numerical flux (see Figure 1). For a first order scheme based on a three points stencil:

ξ i min = min( ξi-1 min , ξi min , ξi+1 min ) ξ i max = max( ξi-1 max , ξi max , ξi+1 max ) (19)
ξ i min and ξ i max are respectively the new minimum and maximum velocity contained in the local grid for cell i. This step ensures that not only the distribution function is zero outside the local grid but the numerical fluxes at the grid interfaces are also zero.

For a second order scheme in space, there are actually two more cells involved in the stencil of the transport step for each cell. Indeed, to calculate the flux between cell i and cell i + 1, one needs the slopes in cell i and in cell i + 1. In the cell i + 1 the slopes are calculated with [START_REF] Filbet | An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation[END_REF] which involves the distribution function in cell i + 2. Thus, the union is performed as follows:

ξ i min = min( ξi-2 min , ξi-1 min , ξi min , ξi+1 min , ξi+2 min ) ξ i max = max( ξi-2 max ξi-1 max , ξi max , ξi+1 max , ξi+2 max ) (20)
With this approach, the maximum microscopic velocity used for the computation of the transport step is not necessarily the maximum velocity of the global grid G. It may also vary in time but it remains at most the maximum velocity of the global grid. To improve the computational time, at each time step a new ∆t can be computed considering the maximum velocity ξ max of all the local velocity grids. This modification may reduce the CPU time in all cases where the maximum velocity of the global grid grid is not contained in any local grid. This approach ensures that for a Maxwellian distribution function and α ≥ 5 the density error is less than 10 -6 . However, the error can be larger for high Knudsen numbers, when the distribution function is not Maxwellian. In order to account for these cases, the local grid at the new time step is chosen as follows. First, in a space cell i at time t n , the local grid G n i is estimated as in [START_REF] Mieussens | Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries[END_REF]. Then a test is performed on the distribution function to check that the new local grid contains a significant part (set with a tolerance) of the distribution function stored in the old grid G n-1 i . This condition reads as follows:

| G n i f n i dξ -G n-1 i f n i dξ| G n-1 i f n i dξ < tol (21)
where tol a tolerance (set to 10 -5 in all numerical test cases). If [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF] does not hold,

G n i = G n-1 i .
Finally, the union of the neighbour velocity grids is performed as in [START_REF] Pareschi | Implicit-Explicit Runge-Kutta Schemes and Applications to Hyperbolic Systems with Relaxation[END_REF].

Defining the local velocity grids in such a way ensures that if all the distribution function is contained in the local grids at the first time step, the convective fluxes outside the local grids are zero even if the distribution function is not a Maxwellian. Let us summarize the algorithm to define the local grid in the space cell i at the beginning of each time step for a 1D case. Let G n-1 i be the grid used in the previous time step:

Inria x x i x i+1 x i-1 ξ ξ j =0 ξ j+1 ξ j+2 ξ j+3 ξ j+4 ξ j+5 ξ j+6 ξ j-1 ξ j-2 ξ j-3 ξ j-4
1. Compute the maximum and minimum velocity required with respect to the parameter α, the macroscopic velocity and the temperature as in [START_REF] Mieussens | Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries[END_REF] 2. Check if the distribution function f n i is contained in the new local grid G n i with respect to a tolerance as in [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF]. Otherwise set

G n i = G n-1 i .
3. Perform the union with the neighbour local velocity grids with (20) 4. Compute the numerical fluxes using the new grid and evolve f n+1 i .

Note that step 2 is not performed at t=0 because the distribution function at t=0 is initialized with the local Maxwellian in each space cell. Thus the criteria on the standard deviation is enough to initialize the local velocity grids with good accuracy.

The multidimensional space algorithm

The extension to multidimensional space is quite straightforward. The same approach is used along all dimensions independently. By considering the local macroscopic velocity and local temperature, criteria ( 18) is applied to construct the local grid in each space direction with the corresponding component of the macroscopic velocity. The maximum and minimum microscopic velocity ( ξmax = ( ξu,max , ξv,max , ξw,max ) and ξmin = ( ξu,min , ξv,min , ξw,min ) in 3D) are then computed along each space direction independently. To make sure of including all non zero flux on non-existing velocities (as described in 1D), a union of the neighbour velocity grids has to be performed. In multidimensional cases, the fluxes are calculated with a directional splitting.

Each local grid has to include all the local grids involved in the transport step. For example, in 2D, in a space cell (i, j) and for a second order scheme in space:

ξ i,
As in the 1D case, this kind of approach might not be accurate away from the hydrodynamic regime. Therefore, the same correction of the local velocity grid is applied based on the integral of the distribution function (Eq.( 21)).

The local grids are then a sub-rectangle of the global grid in 2D and a brick in 3D.

Numerical results

In this section numerical results will be presented in 1D and 2D. The computation for the 2D test cases is done in parallel.

Test case 1: Two interacting blast waves

Hydrodynamic regime

This test case was initially introduced by Woodward and Colella [START_REF] Woodward | The numerical simulation of two-dimensional fluid flow with strong shocks[END_REF]. It is a popular test case that shows the ability of a scheme to correctly capture very strong shock waves in the continuum regime. Initially, the domain [0,1] is decomposed in three sub-domains: In each subdomain, the distribution function is initialized as a Maxwellian. Due to the large temperature differences, the support of each Maxwellian is very different as shown on Figure 2.

   ρ = 1, u = 0, p =
At the boundaries of the domain, solid walls reflect the waves. The classic specular reflection is applied. The Knudsen number is set to 10 -5 and the simulation is stopped at t=0.01. Second order schemes in space and time are used. The global velocity grid is computed as in [START_REF] Kennedy | Additive Runge-Kutta schemes for convectiondiffusion-reaction equations[END_REF]. In this case, the velocity space is [-160,160] and is discretized uniformly with 3200 grid points for the global grid. The solution is shown on Figure 3 for the temperature and the density. The solid line corresponds to the solution of the BGK equation on the global grid, the dashed line corresponds to the solution of compressible Euler equations while the solution for the BGK equation with the local grid method and α = 6 is represented with + signs.

The local velocity grids are defined as in ( 18) and ( 21), and we investigate the performance of the scheme as a function of the parameter α. Note that increasing α means that the local grid boundaries are enlarged. Figure 4 shows the boundaries of the local grids as a function of space when only criteria ( 18) is used with α = 6. Taking into account the second criteria enlarges the local grid around shocks. Comparing it with the global grid (in solid lines) highlights that the local grid includes only a fraction of the global grid points. L ∞ norm on Figure 6. Defining ρ = (ρ, U, E), the different norms are normalized as follows:

L p,normalized (ρ) = ||ρ G -ρ Gv || Lp ||ρ G || Lp ( 23
)
where L p is the standard L p norm, ρ G is the solution for ρ on the global grid, ρ Gv is the solution for ρ with the local grid method.

It is observed that the error decreases fast and for α ≥ 6 the error is already lower than 10 -6 in all norms. It remains stable for large values of α.

One can also compute the conservation error on mass and energy normalized by the initial value of mass and energy respectively (Figure 7) as a function of α. This error has a similar behaviour and reaches quickly machine precision (which corresponds to the conservation error Results on Figure 8 show the fractions of the computational time and of the total number of velocity grid points used with respect to the global grid calculation as function of α. These results are obtained with a constant time step defined with the maximum velocity of the global grid. Even for α = 600 the velocity grid does not coincide with the global grid because of the presence of regions where the temperature is small. Indeed, in such areas the standard deviation of the Maxwellian is small (∼ 0.1) and a large value of α is required to get the whole grid (α ∼ 1600). If the time step is defined at each iteration with the maximum velocity contained in the local grids (but still constant in space), the computational time can be significantly reduced when the maximum velocity of the global grid is not contained in any local grid. Figure 9 shows the comparison of the computational time obtained with a constant time step and a variable time step defined as above.

The largest gain is obtained for small values of α. But even with higher values, there is still a gain. For instance, for α = 6, the error is below 10 -6 with a conservation error below 10 -5 while the gain in CPU time is around 50% using less than 75% of the global grid.

Rarefied regime

Here the method is tested on the same test case but with a higher Knudsen number. In particular, we consider Kn = 10 -2 as an intermediate regime, and Kn = 1 for the fully kinetic regime.

In these cases, the error of the method is larger with respect to the case at low Knudsen number because the distribution function may be more diffusive than the Maxwellian and thus the conservation error may increase as Kn grows. Since the shape of the distribution function is not close to a Maxwellian distribution function, criteria [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF] on the integral of the distribution function has to be used. Hence, the error still goes towards zero for reasonable values of α. We show the results on the error in L ∞ on Figures 11. In other norms, the errors behave similarly. These results also show that for α around 5, there is a kink in the error. This occurs when the conservation criteria [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF] becomes less restrictive than the criteria on α. Typically, for these values of α (around 5) the more restrictive criteria switches from one to another during the calculation. The same behaviour can be observed in the following error curves relative to 2D hydrodynamic or kinetic regime.

The gain in the computational time (see Figure 12) is much smaller than for Kn = 10 -5 because the relaxation time is too high to redistribute the particles as a Maxwellian distribution function after the convective step. Then, after several time steps, the distribution function lies on a large part of the velocity grid in all cells. But a gain of about 15% can be obtained for a relative error of 10 -4 , which is obtained at α = 6.

Test case 2:

A Mach 3 wind tunnel with a step

The hydrodynamic regime

Here, the local grid method is tested on a 2D case. A flow at M = 3 propagates in a tunnel with a step. A shock is created on the step and then is reflected on the top of the tunnel. Strong shocks propagate and interact with themselves and with the boundaries. This is also a difficult test case for the discrete velocity model because of the step geometry. The temperature profile is shown in Figure 13.

The domain is [0,3]×[0,1]. The step lies from x = 0.6 to x = 3 and goes from y = 0 to y = 0.2. The spatial domain is discretized with 150×50 cells. The Knudsen number is set to 10 -5 . The global velocity space is [-20,20]× [-20,20] discretized on a Cartesian grid with 61 grid points in each direction. It is built to satisfy conditions [START_REF] Kennedy | Additive Runge-Kutta schemes for convectiondiffusion-reaction equations[END_REF].

The error is calculated with respect to the BGK solution on the whole velocity grid for different values of α for density, velocity and temperature. The L ∞ norm of this error is plotted on Figures 14. Other norms behave similarly.

As for the 1D case, the error on macroscopic variables goes quickly towards zero as α grows. For values of α between 5 and 10, the error is lower than 10 -4 in all norms. Figure 15 shows the computational time and the total number of velocity grid points used for one iteration on one processor. It shows a good scalability of the method with respect to the number of velocity grid points. For small values of α (≤ 5) the computational time becomes less sensitive to the number of velocity grid points used because the execution time of some parts of the code is independent or almost independent of the number of velocity grid points. For example, the time required to compute the Maxwellian distribution function is mostly due to the solution of the non-linear system in [START_REF] Chen | A unified gas kinetic scheme with moving mesh and velocity space adaptation[END_REF] and the number of space cells.

The computational time to get the solution at t=1.2 has also been compared for constant and variable time steps. Also the total number of velocity grid points used for the whole calculation (see Figure 16). It shows that the method is efficient because the computational time scales well with the number of velocity grid points. With α = 10, less than 30% of the whole grid is used and the computational time is reduced by more than 50% with a good accuracy (the L ∞ norm is lower than 10 -6 for all variables).

We also display the number of velocity grid points used in each cell at the final time (Figure 17). More detail is needed in the regions where there are sharp gradients, and where the gas is far from equilibrium. The width of the grid is determined by the temperature and thus it follows the temperature profile (compare Figure 13 and17).

The rarefied regime

The same test case is presented but with a Knudsen number set to 10 -2 . Here, the distribution function will no longer be close to a Maxwellian. The criteria [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF] on the integral of the distribution function is more restrictive than the criteria involving α especially for small values of α. The results on the error in L ∞ norm are shown in Figure 18 while the computational gains 19. For α = 10, the error is of the order of 10 -5 and the CPU time is again reduced by more than 50%.

Test case 3: the blunt body at Mach 10

Another 2D test case is presented to show the efficiency of the method on a Cartesian grid with immersed boundaries. A cylinder is immersed in a flow at Mach 10. A shock develops on the body and very strong differences of temperature and velocity will appear in the field. We consider Kn = 10 -5 and therefore we impose the Euler-AP boundary condition on the cylinder [START_REF] Bernard | Second-order Euler-preserving wall condition for the BGK equation on Cartesian meshes[END_REF]. This test requires a very large and fine velocity grid to capture all the phenomena. The domain is [-2,0]× [0,4] and is discretized with 50× 100 cells in space. The global velocity space is discretized in 101×101 cells in [-50,50]×[-50,50].

The solution is taken at convergence to steady state and the error obtained with the local grids is calculated with respect to the solution on the global velocity grid (Figure 20).

Taking an error of the order of 10 -6 (see Figures 20) corresponds to α = 6 and gives a gain about 70% of the computational time (see Figure 21).

In Figure 21, one can note that for the smallest value of α (α = 2), the number of grid points used is actually larger than for α = 3. This is due to the criteria on the integral [START_REF] Pieraccini | Implicit-Explicit schemes for BGK kinetic equations[END_REF] which is more restrictive and imposes to enlarge the grid to include all points where f cannot be disregarded. This occurs also in the case of the wind tunnel (see Figure 19).

Conclusion

We presented an approach to solve problems governed by the BGK equation where high temperature or velocity gradients are present in the field. This method is based on the construction of local velocity grids. In each space cell, the local grid is determined by the local macroscopic velocity and temperature. These quantities a-priori determine the shape of the corresponding Maxwellian distribution function. Thus, at each time step, the local grid is defined ensuring that all regions where the local Maxwellian is not negligible are included. Next a criteria on conservation is added to the algorithm to avoid mass and energy loss in the kinetic regime where the distribution function is far from the local Maxwellian. The local grid is characterized by having always the same grid spacing and coincide with the global velocity grid. This ensures that no interpolation is required between neighbouring space cells, avoiding the overhead typical of non uniform grids. Though the scheme is very simple, the computational gain is relevant, especially close to the hydrodynamic regime. These gains are observed also in parallel computations. In future work, this method will be extended to the ES-BGK model to deal with the correct Prandtl number and to 3D test cases where the gain expected is even larger.

Figure 1 :

 1 Figure 1: Representation of velocity-space cells in phase space in 1D. Local grids defined in cell i -1, i, i + 1 before the union are in solid lines and stencil used for the transport in space cell i after the union is in dashed lines. Cells in the local grid for space cell i are hatched. Cells with marker ⋆ are cells added after the union.

Figure 2 :

 2 Figure 2: Test Case 1: Normalized Maxwellian distribution functions in region A (left), B (center) and C (right)

Figure 3 :

 3 Figure 3: Test case 1, Kn = 10 -5 : Density (left) and temperature (right) solution with 500 grid points in space

Figures 5 and 6 Figure 4 :

 64 Figures 5 and 6 show the normalized errors in L 1 , L 2 and L ∞ norm for the density, velocity and energy for several values of α. The behaviour for large values of α is also presented in

Figure 5 :Figure 6 :

 56 Figure 5: Test case 1, Kn = 10 -5 : L 1 (left) and L 2 (right) error on density, velocity and energy

Figure 7 :

 7 Figure 7: Test case 1, Kn = 10 -5 : Conservation error

Figure 8 :

 8 Figure 8: Test case 1, Kn = 10 -5 : Normalized number of velocity grid points used and computational time with respect to the global grid calculation

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: Test case 1, Kn = 10 -5 : Normalized number of velocity grid points used and computational time with respect to the global grid calculation

Figure 12 :

 12 Figure 12: Computational time and total number of velocity grid points used for Kn = 10 -2 (left) and Kn = 1 (right)

Inria

Figure 13 :Figure 14 :

 1314 Figure 13: Solution for the temperature (Kn = 10 -5 )

Figure 15 :

 15 Figure 15: Normalized number of velocity grid point used and computational time with respect to the full grid calculation for one iteration on one processor (Kn = 10 -5 )

Inria

Figure 16 :Figure 17 :Figure 18 :

 161718 Figure 16: Normalized number of velocity grid point used and computational time with respect to the full grid calculation (Kn = 10 -5 )

Inria

  number of velocity grid points

Figure 19 :

 19 Figure 19: Test case 2, Kn = 10 -2 : Normalized number of velocity grid points used and computational time with respect to the full grid calculation

Figure 20 :

 20 Figure 20: Test case 3, Kn = 10 -5 : L ∞ error on density, velocity and temperature

Figure 21 :

 21 Figure 21: Test case 3, Kn = 10 -5 : Normalized number of velocity grid points used and computational time with respect to the full grid calculation

RR n°8472

avenue de la Vieille Tour 33405 Talence Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399


Alexis Michaud

Marc Brunelle

Chapter 40: Information Structure in Asia: Yongning Na (Sino-Tibetan) and Vietnamese (Austroasiatic)

Keywords: information structure, Yongning Na, Vietnamese, referential density, intonation, discourse particles

Languages of Asia are highly diverse. Rather than attempting a review of the literature about information structure in this huge linguistic area, this chapter provides observations about two languages that differ sharply in terms of how they convey information structure. Yongning Na (Sino-Tibetan) is an example of a language with abundant morphemes expressing information structure, whereas Vietnamese (Austroasiatic) makes little use of such morphemes. Along with key morphosyntactic facts, this chapter presents the role played by intonation in conveying information structure in these two tonal languages.

Introduction

The present chapter is intended to be complementary with the overview of information structure (hereafter IS) in Chinese by Chen and Pan (this volume). Neither of the two languages presented in this chapter, Yongning Na and Hanoi Vietnamese, belongs to the small set of "canonically exotic languages" (Matisoff 1973:xlv), such as Japanese, which are often discussed in the literature, so their inclusion in a textbook can be seen as a modest contribution towards broadening the empirical basis of IS research. Moreover, these two languages differ sharply in the way they convey IS, providing an illustration of the great diversity found among languages of Asia.

"Information structure is a vast topic of research that has been pursued within different theoretical frameworks" (Krifka 2008:244), and with different objectives in view. While we have adhered to the framework set by [START_REF] Krifka | Basic notions of information structure[END_REF], we are keenly aware that cross-linguistic categories can only capture a small part of IS (see [START_REF] Matić | The meanings of focus: the significance of an interpretation-based category in cross-linguistic analysis[END_REF], and the general discussion by [START_REF] Haspelmath | Pre-established categories don"t exist: consequences for language description and typology[END_REF]; we have accordingly stated our observations in terms that we hope can easily be interpreted by readers working within different theoretical frameworks. Our view of IS is that it is not to be viewed as a separate "module", but as part and parcel of language as communicative activity. IS "cannot be described without referring to the strategies used to narrate events or make arguments" (Krifka 2008:272; see also [START_REF] Kohler | Patterns of prosody in the expression of the speaker and the appeal to the listener[END_REF]Horn, this volume;Karcevskij 1931:191).

Each of the two sections of this chapter deals successively with topics, focus and givenness.

Yongning Na

Yongning Na is a language of the Naish subgroup of Sino-Tibetan [START_REF] Jacques | Approaching the historical phonology of three highly eroded Sino-Tibetan languages: Naxi, Na and Laze[END_REF]. It has typologically salient characteristics, many of which are shared with other Sino-Tibetan languages.

Topics

Word order

The following generalization about Qiang proposed by [START_REF] Lapolla | A grammar of Qiang with annotated texts and glossary[END_REF] also applies to Yongning Na: "The structure of the clause is to some extent affected by pragmatic factors, but this only applies to the order of noun phrases in the clause. The utterance-initial position is the unmarked topic position (though secondary topics can follow the primary topic), while the position immediately before the verb is the unmarked focus position, and so the focused element will generally appear there. The verb always appears in final position; there is no possibility for the actor of a clause to appear in postverbal position, even if it is focal. The only exception to this is the occasional afterthought clarification of a noun phrase that was omitted or expressed as a pronoun in the clause." (LaPolla and Huang 2003:221) IS in Na has accordingly been described as "topic-comment", extending an observation made by Chao Yuen-ren about Chinese: "the grammatical meaning of subject and predicate in a Chinese sentence is topic and comment, rather than actor and action" (Chao Yuen-ren 1968:69;[START_REF] Shi | Topic and topic-comment constructions in Mandarin Chinese[END_REF][START_REF] Lapolla | Chinese as a Topic-Comment (not Topic-Prominent and not SVO) language[END_REF].

"The primary information structure in Na is topic/comment rather than subject/predicate. (...) a topic can be a nominal argument, about which the rest of the sentence will comment upon, but the topic can also be an adverbial, an independent clause, or a dependent clause." (Lidz 2010:296) Word order thus plays an important role in the structuring of IS in Na. Example (1) provides an illustration. 1 Sentence (1) is translated with a conditional, illustrating the proximity between topics and conditionals [START_REF] Haiman | Conditionals are topics[END_REF]).

1 Unless specified otherwise, the examples provided in the chapter have been collected by the authors or coined and double-checked with native speakers. Apart from example (1), all the Yongning Na examples are drawn from a set of 20 narratives; the recordings are available online with time-aligned transcriptions through the Pangloss Collection [START_REF] Michailovsky | Documenting and researching endangered languages: the Pangloss Collection[END_REF]. The reference to these texts is provided in the following format: <short title of document> <version number, for narratives that were told several times>.<sentence number>.

For instance, Housebuilding.259 refers to sentence 259 of the narrative "Housebuilding", and Seeds2.67 refers to sentence 67 in the second version of the narrative "Seeds". The numbering of sentences is indicated in the online display of the texts. Glossing follows the Leipzig Glossing Rules, with the following additions to the core set of standard abbreviations: ACCOMP for ACCOMPLISHED; ADVB for ADVERBIALIZER; CERTITUDE for an epistemic value of the copula in Yongning Na; and EXIST for EXISTENTIAL verbs. Low, and the combinations for ow-to-High and for Mid-to-High; for a presentation of the Yongning Na tone system, see [START_REF] Michaud | The tone patterns of numeral-plus-classifier phrases in Yongning Na: a synchronic description and analysis[END_REF].

relations). In (1), the topic marker /-dʑo / does not appear; a plausible reason is that in this context (after a verb preceded by the accomplished morpheme /le -/) /-dʑo / would be interpreted as the progressive, which is homophonous with the topic marker. In the second sentence in (2), on the other hand, the topic marker /-dʑo / appears as expected:

(2) kʰv m -ʈʂʰɯ -ɳɯ , | h -bv , | ə tso do , | ə tso ʈʂ -ɲi -ze ! | q do do -dʑo , | q do q ! kʰv m -ʈʂʰɯ -ɳɯ h -bv ə tso do ə tso bandit In the first sentence in (2), "robbers" is strongly brought to the fore through the combination of /-ʈʂʰɯ / and /-ɳɯ/. These two particles contribute to indicate that the preceding noun has the status of topic; on the other hand, /-ɳɯ/ and /-ʈʂʰɯ / are often associated with the topic marker /-dʑo /, highlighting the fact that they are not yet fully grammaticalized as topic markers. The marker /-ʈʂʰɯ / was grammaticalized from a proximal demonstrative, also used as 3 rd -person pronoun, with which it remains formally identical. The postposition /-ɳɯ/ is an ablative marker, which has developed into an agent marker (Lidz 2011:53; for observations on similar developments in neighbouring languages: LaPolla 1995; on the use of case markers to convey IS, see also the alternation between ergative and absolutive in Newari: [START_REF] Genetti | A syntactic correlate of topicality in Newari narrative[END_REF]. A close examination of cases of use of /-ɳɯ/ as an agent marker reveal that it is fairly rare for there to be true ambiguity as to semantic roles, however (Lidz 2011:55). Cases like (2) are common:

the agent marker functions prominently as an IS indication.

From a lexicographic point of view, it is an issue whether to set up a distinct subentry It appears appropriate to set up three subentries for the morpheme /-ʈʂʰɯ /, since its uses as proximal demonstrative, as third person pronoun and as topic marker are by now relatively well-differentiated. The three glosses used are DEM.PROX, 3SG and TOP(/DEM.PROX): the latter combines an indication of the particle"s IS function (topic marker) with an indication of its origin as a proximal demonstrative. On the other hand, the use of /-ɳɯ/ as IS marker does not currently appear highly grammaticalized; omission of this particle from (2) would not modify IS in any spectacular way, even though it would lend less salience to the topic. It therefore does not appear warranted to set up of a distinct subentry; the three uses of /-ɳɯ/ -as ablative, agent and topic marker -are therefore indicated side by side in the gloss, in the hypothesized order of diachronic development: ABL/A/TOP.

In addition to /-dʑo /, /-ʈʂʰɯ / and /-ɳɯ/, there exist three less frequently occurring topic markers, whose origin remains unclear. One is an archaic word, /-l /, appearing only once in the recorded data. The two others are /-no / and /-se/, which can only apply to a noun phrase, whereas the other topic markers can bear on a noun phrase or an entire clause. Examination of the contexts of appearance of /-no / and /-se/ brings out the presence of strong emphasis on the topicalized element in contrast to others: a possible translation is "as for X/as concerns X".

These two morphemes can be described as contrastive topic markers (Krifka 2008:267).

Focus

Focus in Yongning Na is conveyed by local phenomena of intonational emphasis. The discussion progresses from emphatic stress to less extreme cases of intonational emphasis.

Emphatic stress

Emphatic stress is found in Yongning Na. It appears to have essentially the same function as in English and French -calling the attention of the listener to a particular word -hence the choice to use this label (proposed by [START_REF] Coustenoble | Studies in French intonation[END_REF]. In the languages where emphatic stress has been experimentally investigated, it has been found that prototypical realizations involve supplementary activity of the expiratory muscles, resulting in a sudden increase in subglottal pressure during the articulation of a consonant [START_REF] Benguerel | Corrélats physiologiques de l"accent en français[END_REF][START_REF] Carton | L'accent d'insistance[END_REF][START_REF] Ohala | The Production of Tone.' Tone: a linguistic survey[END_REF][START_REF] Fant | Focal accent and subglottal pressure[END_REF], hence the term "force-accent" used by [START_REF] Kohler | Neglected Categories in the Modelling of Prosody: Pitch Timing and Non-Pitch Accents[END_REF]. Acoustically, this exerts an influence on spectral slope (Rossi 1971:143;[START_REF] Glave | Is the effort dependence of speech loudness explicable on the basis of acoustic cues?[END_REF][START_REF] Gobl | Voice source dynamics in connected speech[END_REF][START_REF] Fant | The voice source in prosody[END_REF][START_REF] Sluijter | Spectral balance as a cue in the perception of linguistic stress[END_REF][START_REF] Heldner | On the reliability of overall intensity and spectral emphasis as acoustic correlates of focal accents in Swedish[END_REF].

Emphatic stress has been somewhat neglected in intonation studies, as researchers focused their attention mostly on the acoustic parameter of fundamental frequency. But it is an important component of intonation, in Na as well as in a number of other languages. Its linguistic functions range from the attitudinal and emotional to the pragmatic. It is most often encountered in toned-down versions, physiological effort at a subglottal level being mimicked through such strategies as f0 excursions and consonant lengthening. Like other linguistic phenomena, emphatic stress comprises important language-specific and speech-style-specific dimensions: its frequency of use varies greatly from language to language, from speaker to speaker, and from style to style; its stylistic effect is inversely proportional to its frequency of use.

An "up" arrow ↑ to mark intonational emphasis is adopted for the Na data, following [START_REF] Mazaudon | Language Variation: Papers on variation and change in the Sinosphere and in the Indosphere in honour of[END_REF]. The arrow is placed to the left of the emphasized syllable, as in (3).

(3) tʰi , ə mv -ʝi -h | -dʑo , ↑zo ɲi -tsɯ | -mv . In many contexts, emphatic stress appears on a constituent that can be predicted to receive normal focus prosody. For example, in (3), we would expect the focus to be in the immediate preverbal position -the usual unmarked focus position for verb-final languages. Emphatic stress can be considered an extreme form of focus prosody; it is an extreme along a continuum: there is no hard-and-fast boundary between emphatic stress and milder realizations of focus prosody. When annotating recordings, it is sometimes an issue whether to add an "up" arrow ↑ or not.

The phonetic realization of emphatic stress includes effects on the articulation of vowels and consonants also partakes in the realization of emphatic stress: for instance, the second syllable of the verb /dʑɤ ↑bv / "to play" is realized in Caravans.231 with a much stronger trilling of the /b/ than is found in non-emphatic contexts. (About "articulatory prosodies", see [START_REF] Kohler | On the role of articulatory prosodies in German message decoding[END_REF][START_REF] Niebuhr | Intonation segments and segmental intonations[END_REF][START_REF] Niebuhr | On the acoustic complexity of intonation[END_REF] Emphatic stress is phonetically located on one syllable only, but from the point of view of interpretation, there is ambiguity of focus marking. This phenomenon is extensively studied in the literature on focus projection (e.g.

Selkirk 1995); Lambrecht uses the terms "broad focus" vs. "narrow focus" [START_REF] Lambrecht | Information structure and sentence form[END_REF].

In its most vehement manifestations, emphatic stress intrudes into a sentence"s intonation, wreaking havoc on tonal contrasts. Example ( 4) is a case in point.

( In Yongning Na, one means of emphasizing a word within a sentence consists in a rapid dipping contour, as in example (5). Focalization through a dipping contour is compatible with emphatic stress, but formally distinct from it.

(5) h -ki | ɖɯ -kʰwɤ F | mə -pi h ki ɖɯ -kʰwɤ mə pi person to 1-CLF_pieces NEG say "(S)he did not say anything to the people present! / (S)he did not greet anyone!" (Field notes,

2009)

The phenomenon marked as "F", for "intonational Focalization", is an intonational device to Yongning Na does not require pronouns to stand in place of the pragmatically inferable noun phrases, as illustrated by (1); this can be referred to as a case of "Pro-Drop" (for Pronoun Dropping; Chomsky 1981), but as pointed out by Launey (1994:45, 93), the perspective could just as well be reversed, considering languages that require overt NP arguments as a special case (widely represented in Western Europe: Haspelmath 2001).

There is a limit to economy in referential density, however. For instance, in example

(2), one could be tempted to push referential economy further, and -in addition to ellipsis of the subject argument -to avoid the repetition of the object /q do / "lumber". However, it is not acceptable to remove the second /q do / "lumber": when this manipulation was attempted, the result, shown in (2"), was rejected by the consultant.

(2") *q do do -dʑo , | q q do do -dʑo q lumber see TOP burn Intended meaning: "If they spot lumber, they will burn it!" (adapted from Housebuilding.259)

This probably has to do with several factors. First, there exist numerous homophonous roots, following monosyllabicization and phonological erosion [START_REF] Michaud | Monosyllabicization: patterns of evolution in Asian languages.' Monosyllables: from phonology to typology[END_REF]; lexical identification can be difficult for a monosyllabic verb without an object. A second factor that may be at play is rhythm and the overall symmetry of the utterance (what see, what rob; lumber see, lumber rob). A third factor, certainly not the least important, concerns IS: "(they) rob lumber" is the focal information, so it would be inappropriate to give it a less complete representation than other components of the utterance. Cross-linguistically, there is a closer syntactic integration of object and verb, as compared with subject and verb, and objects are more likely than subjects; this goes a long way towards explaining why all languages that allow Object Pro-Drop also allow Subject Pro-Drop, whereas the reverse is not true.

How prosodic phrasing reflects information structure: the division of the utterance into tone groups

A salient characteristic of Yongning Na is the interaction of IS and morphotonology.

Yongning Na possesses abundant tonal morphology, comparable in its extent to the segmental morphology found in the (very distantly related) Kiranti languages [START_REF] Jacques | An overview of Khaling verbal morphology[END_REF][START_REF] Jacques | Agreement morphology: the case of Rgyalrongic and Kiranti[END_REF]). For instance, the determinative compound /ʐw zo -gv dv / "colt"s back" has a final H tone, whereas the coordinative compound "father and mother", which has the same input tones, is /ə d -ə mi /, with a floating H tone (a tone which can only associate to a following syllable; it is represented as ). The floating H tone on "father and mother" is not by itself a marker of the morphological status of the compound as being coordinative rather than determinative: not all determinative compounds share the same tone pattern, any more than coordinative compounds do. The output tone depends on the input tones, but different rules apply in determinative compounds and in coordinative compounds. The tone rules that apply when combining an object with a verb are likewise different from those that apply when combining a subject with a verb. A systematic study of numeral-plus-classifier determiners brings out no less than nine categories with different tone patterns [START_REF] Michaud | The tone patterns of numeral-plus-classifier phrases in Yongning Na: a synchronic description and analysis[END_REF]. The tone group is the domain within which these abundant morpho-phonological tone rules apply: each utterance consists of one or more tone groups; tonal computation takes place separately in each tone group. The phenomena that refer to this domain are exclusively tonal: they consist of tone rules, in particular the following seven rules: (i) tone spreads progressively ("left-toright") onto syllables that are unspecified for tone, (ii) syllables that remain unspecified for tone after the application of Rule 1 receive M tone, (iii) H and M are neutralized to M in tonegroup-initial position, (iv) a syllable following a H-tone syllable receives L tone, (v) all syllables following a HL or ML sequence receive L tone, (vi) LH and LM contours on a tonegroup-final syllable are neutralized to LH, and (vii) if a tone group only contains L tones, a post-lexical H tone is added to its last syllable.

As illustrated by the examples below, different divisions into tone groups have different implications in terms of prominence of the various components. Tone groups may be considered as constituting one phonological phrase each; changes in phonological phrasing serve to highlight a certain element, lending it focal prominence. (The term "tone group" is nonetheless used here in preference to "prosodic phrase", as the defining characteristic of this phonological unit is that it serves as the domain of tonal processes.)

A tone group boundary is always found after topicalized phrases; but apart from this hard-and-fast rule, speakers generally have several options for dividing the utterance into tone groups. They may choose to integrate large chunks of speech into a single tone group, resulting in a stronger integration; or they may divide the utterance into a number of tone groups, with the stylistic effect of emphasizing these individual components one after the other. The latter option is illustrated by ( 6):

(6) dzɯ -di | mə -dʑo , | ʈʰɯ -di | mə -dʑo dzɯ -di mə dʑo ʈʰɯ b -di mə dʑo eat NMLZ NEG EXIST drink NMLZ NEG EXIST
"[Before mankind had learnt to grow crops], there was nothing to eat and nothing to drink!" (Seeds2.67)

The noun phrase /dzɯ -di / "food" and the negated existential verb /mə -dʑo / "there isn"t" can be separated into two tone groups, as in ( 6). This has the effect of emphasizing the two noun phrases, /dzɯ -di / "food; thing to eat" and /ʈʰɯ -di / "drink; beverage". The following sentence in the story repeats the statement "There was no food", continuing the same strategy of bringing out the noun phrase "food", this time with the topic marker /-dʑo /: /dzɯ -di |dʑo , | mə -dʑo -ɲi -tsɯ | -mv / "As for food, it"s said that there was none!" (Seeds2.68).

Then the narrator recapitulates, and moves on:

(7) dzɯ -di mə -dʑo | -dʑo | tʰi dzɯ -di mə dʑo -dʑo tʰi eat NMLZ NEG EXIST TOP
so/then "As there was nothing to eat, ..." (the narrative moves on to: "there were some exceptional, smart people, who stood up and did something about it") (Seeds2.69)

At this juncture, "there was no food" is integrated into a single tone group, and followed by the topic marker /-dʑo /. This provides an exemplary illustration of the integration of larger chunks of information into a single tone group as this information changes its status from new to old and backgrounded.

Conversely, insertion of a tone-group boundary highlights the word or phrase that precedes. Even function words can be emphasized in this way. Consider ( 8): A simpler formulation would be /ɬo pv -ti -kv -tsɯ -mv /, integrating the passage into a single tone group. The formulation in ( 8) emphasizes the reported-speech particle. This particle is used over and again by the consultant when telling narratives: it is used whenever the speaker only has indirect knowledge of the situation at issue. But in the context of ( 8), the emphasis laid on this particle is one of the manifestations of the speaker"s efforts to adhere to truthfulness and precision: the narrator never witnessed the ritual that she describes. The particle could be paraphrased, in this context, as: "at least, that"s what they say", or "but I can"t personally vouch for it: this is just hearsay, you know".

As a final example, consider ( 9 In ( 9), the noun phrase "dog meat" is set into relief by constituting a tone group on its own.

Despite the absence of a morphemic indication that it is topicalized, it clearly has the status of topic. In this context, tonal integration with a following verb would not be stylistically appropriate.

This is parallel to prosodic grouping in English: the phrase "the Massachusetts Institute of Technology" contains three lexical stresses, but someone who is familiar with the place is likely to pronounce it (when not using the acronym "MIT") as one single prosodic phrase, Like Yongning Na,Vietnamese tends to favour a topic+comment order (Thompson 1965, chapter 10; note that what the author calls "focal elements" corresponds to Krifka"s "topic"; see also [START_REF] Cao | Some preliminaries to the syntactic analysis of the Vietnamese sentence[END_REF][START_REF] Clark | Conjunction as topicalizer in Vietnamese[END_REF][START_REF] Clark | Conjunction as copula in Vietnamese[END_REF]. The relevance of word order for topic marking can be illustrated by a variety of passive-like structures. As these structures are described in more detail elsewhere [START_REF] Simpson | The Vietnamese noun phrase[END_REF], a single example is given in (10). While in (10a)

police is the topic, in (10b) it is Hoa that is topicalized.

(10) a. Another frequent topicalization strategy is pause insertion, as in ( 11). This strategy is not necessarily associated with a special focus, although such an interpretation is not ruled out if focus is marked intonationally (see §3.2). A sentence such as (11) could not be expressed in Na without using information-structure morphemes:

(11) Măng cụt, ai cũng thích.

măng cụt ____ ai cũng thích mangosteen TOP who also like "Everybody likes mangosteens".

A strategy very similar to pause insertion, if more rare, is the use of the equative copula là to mark a neutral type of topic in sentences like (12). Here, the fact that everybody likes mangosteens is merely indexed under the entry mangosteen in the common ground, without any special emphasis. Once again, focus could be intonationally assigned to the topic, but this is not a common strategy. who also like "Everybody likes mangosteens".

More interestingly, Vietnamese has a contrastive topic marker, thì, which, as in ( 13), serves a role comparable to the topic marker /-dʑo / in Na. This topic marker has the effect of contrasting the topic with its alternatives, which [START_REF] Clark | Conjunction as topicalizer in Vietnamese[END_REF] analyzed as an attributive meaning.

(13) Măng cụt thì ai cũng thích.

măng cụt thì ai cũng thích mangosteen TOP who also like "Everybody likes mangosteens (as opposed to other fruits)".

Thì can also be used after an entire sentence, but in that case, it does not clearly serve to rule out alternative interpretations. In ( 14), for instance, thì is used to set an information as a common ground explaining the following clause (and is glossed as "so"). As pointed out by Cao (1992:141), there does not appear to be a clear cut off point between the contrastive and non-contrastive uses of thì.

( "Yesterday, Phương went to the market, so today she stays at home".

When the topic is co-referential with the subject, the use of an anaphoric pronoun in the subject position can also be used to focus it. This is illustrated in (15): Other markers can be used for contrastive topicalization in Vietnamese, such as còn "as for,

and lại "on the other hand". As their English equivalents, these two expressions are only possible if alternatives have explicitly been set into the common ground in previous clauses.

Vietnamese sentences can have multiple topics as in ( 16). While we would not analyse là as a topic/theme marker, but rather as an equative/identification copula (contrary to Cao (Modified from Cao 1992:147).

Focus

Experimental work on the phonetic realization of information structure in Vietnamese is limited to the realization of pragmatic focus [START_REF] Jannedy | Prosodic focus in Vietnamese[END_REF] or corrective focus [START_REF] Michaud | Glottalized and Nonglottalized Tones under Emphasis: Open Quotient Curves Remain Stable, F0 Curve is Modified[END_REF][START_REF] Michaud | Prosodie de langues à tons (naxi et vietnamien), prosodie de l'anglais : éclairages croisés[END_REF]. These studies reveal that these two forms of in-situ focus are realized by most, but not all, speakers as an f0 increase on the target constituent [START_REF] Michaud | Prosodie de langues à tons (naxi et vietnamien), prosodie de l'anglais : éclairages croisés[END_REF][START_REF] Jannedy | Prosodic focus in Vietnamese[END_REF]) and a hyperarticulation of voice quality [START_REF] Michaud | Prosodie de langues à tons (naxi et vietnamien), prosodie de l'anglais : éclairages croisés[END_REF]. Duration and intensity are even more speaker-specific. From this, we can conclude that Vietnamese does not have a single grammaticalized means of marking contrastive focus, but that speakers pick and choose from a pool of hyperarticulation strategies. [START_REF] Jannedy | Prosodic focus in Vietnamese[END_REF] also shows that although listeners can identify focused constituents, their performance is generally poor, and even more so for broad scope focus (entirely VP or entire sentence).

The only other explicit proposal for a phonetic marking of IS in Vietnamese is [START_REF] Thompson | A Vietnamese Reference Grammar[END_REF]. In section 12.5, Thompson describes a three-way stress system in which:

"By far the great majority of syllables are accompanied by medium stress; it is deviation from this in either direction which marks a particular form as conveying an especial heavy or light load in conveying information. In general, weak stress signals information already known or obvious in the context; heavy stress signals new or contrastive information." (Thompson 1965:287) More details are given in (i) Grammatical words seem to have reduced phonetic prominence, while lexical words do not.

(ii) As discussed above, contrastive focus seems to be marked by means of increased prominence. The existence of a prosodically-marked new information focus (phonetically identical or distinct from contrastive focus) is less clear.

(iii) Like many languages [START_REF] Beckman | Lengthening and shortening and the nature of prosodic constituency[END_REF][START_REF] Turk | Word-boundary-related duration patterns in English[END_REF], Vietnamese shows phrase-final lengthening. The nature of the phrasal domains that condition this lengthening is still unclear, as is the exact amount of phonetic lengthening involved. In any case, it is likely that this final-lengthening effect provides cues to prosodic phrasing, which in turn play a role in conveying of IS (see our examples of pause insertion in §3.1).

A systematic description of the intonational marking of focus thus remains to be undertaken.

However, besides in-situ intonational focus, two other strategies are used to mark pragmatic focus. The first one is the use of contrastive topicalization (with marker thì or of a topic followed by a resumptive pronoun), already described in §3.1. The second is the use of cái as a focus marker [START_REF] Nguyễn | The structure of the Vietnamese noun phrase[END_REF][START_REF] Nguyễn | The Vietnamese noun phrase[END_REF][START_REF] Simpson | The Vietnamese noun phrase[END_REF]. Although cái is normally used as a classifier, it also serves as a focus marker, and as such, always precedes a classifier or a mensural noun. Although cái usually conveys the idea of a negative emphasis, as in (17), [START_REF] Nguyễn | The structure of the Vietnamese noun phrase[END_REF] Vietnamese also has focus-sensitive particles like thậm chí/đến "even", cả "also" and chỉ/mỗi "only" that indicate semantic focus [START_REF] Hole | Focus particles and related entities in Vietnamese[END_REF]. Each of these focus-sensitive particles is associated to another particle introducing information that is already part of the common ground. In the case of thậm chí, đến "even" and cả "also", the cũng particle is mandatory, as in (18). In the case of chỉ "only", the particle mới is optional, as in (19). A full discussion of these focus-sensitive particles can be found in [START_REF] Hole | Focus particles and related entities in Vietnamese[END_REF], from which we take up the notion of background particle, which we label BG. Another focus-sensitive particle with a more limited scope is sentence-final không "only" [START_REF] Brunelle | Intonation in Northern Vietnamese[END_REF].

( Illustrative examples of subject and object pro-drop are given in ( 20) and ( 21). As in Yongning Na, the active referent is usually inferred from previous sentences, but it is not uncommon to find sentences in which the referent is inferred from common situational knowledge. For instance, in (20) the subject could be interpreted as "we" or "I", depending on context. Argument ellipsis is often used as a strategy for avoiding an overt argument, especially in contexts where the proper use of pronouns is difficult to assess -for instance when the relative age or hierarchical status of interlocutors is unclear. Inversely, it tends to be avoided in formal situations, especially in the case of pronouns or address terms.

( As in Yongning Na, leaving out an argument is a way of backgrounding it; probably for the same reasons as in Yongning Na, object ellipsis seems rarer than subject ellipsis.

) 20

General conclusion

The above overview offers elements of answer to the question of the relative roles of morphosyntax and intonation in conveying IS. In Na, which has a host of particles conveying IS, intonation plays a lesser role in conveying IS, even though Na has emphatic stress as well as a specific intonational device for focalization. In Vietnamese, intonation seems to play a more important role, although experimental findings reported so far in the literature are largely limited to focus, and cover a relatively narrow range of speaking styles. The nature of the language"s tone system also appears to play a role. The multiplicity of phonetic cues to tone in Vietnamese allows for greater malleability: for instance, a tone that has final glottalization can have its overall F0 register raised to convey intonational information [START_REF] Michaud | Glottalized and Nonglottalized Tones under Emphasis: Open Quotient Curves Remain Stable, F0 Curve is Modified[END_REF][START_REF] Hạ | Modelling the interaction of intonation and lexical tone in Vietnamese[END_REF] without a major threat to lexical identification, so long as the phonation-type characteristics that are part of the definition of this tone are present. On the other hand, in a language where tones are specified solely in terms of pitch levels, such as Yongning Na, intonational modification of the F0 curve is more difficult for the hearer to tease apart from the effect of the tones themselves. On a methodological note, the findings reported above suggest that it is useful to maintain a clear functional distinction between lexical tone, on the one hand, and intonational modifications (reflecting boundaries/junctures and IS), on the other. A full-fledged study of the respective contributions of the various components of prosody (lexical tone; phrasing; prominence; speaker attitude...) in shaping F0 curves remains a task for the future. Research of this type has been attempted for Qiang, a Sino-Tibetan language [START_REF] Evans | Linguistic and human effects on F0 in a tonal dialect of Qiang[END_REF]); general proposals have been put forward for languages with lexical stress [START_REF] Mishra | Decomposition of pitch curves in the general superpositional intonation model[END_REF].

  you] eat [of it], [your] stomach[will] hurt!" (Field notes, speaker F4. Isolated sentence transcribed on the fly. Context: on the mountain, pointing out a berry that is not edible. ) 40.2.1.2. Discourse particles conveying IS Yongning Na is similar to a number of other Sino-Tibetan languages in possessing a wealth of discourse particles that contribute to the structuring of information. Pinpointing the exact range of uses of particles can be a thorny task, given their various connotations and rich compositional potential. Grammars that address these issues in detail are available for Lahu[START_REF] Matisoff | The Grammar of Lahu[END_REF], Lalo[START_REF] Björverud | A Grammar of Lalo[END_REF] Qiang (LaPolla and[START_REF] Lapolla | A grammar of Qiang with annotated texts and glossary[END_REF]. Particles stand at different points along the path towards use as "pure" information-structure markers. In Na, a few particles are exclusively used for the marking of IS, such as the topic marker /-dʑo /; others are about as commonly used as (bleached) information-structure markers and in another function, such as demonstratives; others still are used secondarily to indicate IS, in particular particles indicating the relationship that a noun phrase bears to a verb (thematic2 The vertical bar | indicates a juncture between tone groups; about the division of utterances into tone groups in Yongning Na, see §xyz. Tone is indicated by means of International Phonetic Alphabet tone letters placed after the syllable at issue: for High, for Mid, for

  elder [of the two siblings] was a boy."(Sister.5)

  set part of the utterance into relief. It is realized through a dip in fundamental frequency, accompanied by lengthening and formant movement towards a central vowel. The realization of focalization is sufficiently specific -involving a movement in fundamental frequency, and a change in the vowel: a difference in the time course of formant frequencies -to avoid interference with lexical tone. Emphatic stress is likewise identifiable as such, from cues other than fundamental frequency. This greatly limits the possibility of a misperception of lexical tone caused by these intonational phenomena. (An example is found in Healing.55.)40.2.3 Givenness40.2.3.1. Referential densityOne of the means for backgrounding information in Yongning Na consists simply in not making any mention of it. This holds true of any language, but is especially salient in Yongning Na. Referential density -the ratio of overt to possible argument NPs[START_REF] Bickel | Referential density in discourse and syntactic typology[END_REF] -is low in many languages of the Tibeto-Burman area. The active referent, known from the earlier context, serves as given, without being mentioned(Mazaudon 2003 on Tamang).

  said that [on that occasion, the whole family] will kow-tow!" (Sister3.138)

  integrating the three stressed words and two grammatical words into a single fundamental frequency contour. (On similar phenomena in French, see, among other descriptions,[START_REF] Vaissière | Further note on French prosody[END_REF][START_REF] Rossi | L'intonation, le système du français: description et modélisation[END_REF][START_REF] Vaissière | Prosodic constituents in French: a datadriven approach[END_REF][START_REF] Martin | Intonation du français[END_REF]

( 12 )

 12 Măng cụt là ai cũng thích.

  "t know how to soothe their children" (as opposed to the Vietnamese).

  of Yongning Na. At present, no truly decisive evidence on this issue can be offered; if the speakers could write their language, it would be interesting to see which solution they prefer, but so far there has not yet been any opportunity to work with a

	consultant who had metalinguistic awareness. This phenomenon is provisionally analyzed as
	involving a categorical tone change, from a L.L.H sequence, /pʰv -tɕ ɻ /, to a L.H.L
	4) pʰv -tɕ ɻ -gv -kv -ze -m sequence, /pʰv -tɕ ɻ /.
	pʰv -tɕ ɻ	gv	kv	-ze	-mae
	very_white prepare	ABILITIVE ACCOMP	AFFIRM
	40.2.2.2 Focalization through a dipping contour

"[after boiling, linen thread] can become really white!" (FoodShortage.73)

The usual pronunciation is /pʰv -tɕ ɻ / "very white". In (4), the second syllable is realized phonetically with extremely high fundamental frequency on the syllable /tɕ /, which is considerably lengthened. From a phonetic point of view, its phonetic L tone is conspicuously disregarded. One way of looking at this modification would be to describe it as due to an intonational overlay: functionally, one could consider transcribing as /pʰv -↑tɕ ɻ -gv /, where the arrow ↑ indicates emphatic stress on the second syllable, and the underlying tonal string is unchanged. The lowering of the syllable string /ɻae-gv-.../ would then be ascribed to post-focal pitch range compression.

Another interpretation is that this forcible intonational modification interacts with the phonological tone string of the tone group (on this notion: see section 2.3.2). If the modification of the second syllable in /pʰv -tɕ ɻ -gv / only took place on an intonational level, and the underlying tonal string remained unchanged, the third syllable would retain its phonological H tone. Discussions with the consultant who produced this story suggest that the third and fourth syllables in (4) are lowered to L, however, yielding /...ɻ -gv /, as expected if the second syllable carried H tone: the lowering of all tones to L after a H tone is an exceptionless tone rule

  Cảnh sát giao thông bắt Hoa gần cầu Long Biên.

	cảnh sát	giao thông	bắt	Hoa gần cầu	Long Biên
	police	transportation catch Hoa near bridge	Long Biên
	"The traffic police caught Hoa near ong Biên bridge."
	b. Hoa bị cảnh sát giao thông bắt gần cầu Long Biên.
	Hoa bị	cảnh sát	giao thông	bắt gần cầu	Long Biên
	Hoa NEG.EXP police	transportation catch near bridge	Long Biên

"Hoa was caught by the traffic police near ong Biên bridge."

  1992, but following Clark 1996), the first two constituents are nonetheless both topicalized.

	(16) Ở Tokyo thì máy này mỗi cái là năm ngàn.
	ở	Tokyo thì máy	này ___	mỗi cái là	năm ngàn
	In	Tokyo TOP machine	this TOP	each CLF COP five thousand
	"In Tokyo (as opposed to other places), these machines are five thousand each".

  Thompson"s following subsections (12.51 and 12.52), but he does not explicitly address the issue of the phonetic nature of stress in Vietnamese, and many (if not most) of his examples of stress notation fail to coincide with native speaker"s intuitions (regardless of the dialect). In the end, a close look at Thompson"s examples suggest that he is conflating several types of prominence:

  also gives examples devoid of this negative connotation.

	(17) Cái con chó của hàng xóm sủa suốt đêm thì cả nhà mất ngủ.
	cái con chó của hàng xóm	sủa suốt đêm thì cả nhà	mất	ngủ
	TOP CLF dog PASS neighbour	bark entire night so all house lose	sleep

"The neighbour"s damn dog barked all night so the whole family lost sleep".

)

  Cả Hàn Quốc cũng ăn thịt chó.

	(19) Mỗi người Việt (mới) ăn thịt nhím.
	mỗi	người Việt	mới	ăn	thịt nhím
	FOC_ONLY CLF	Vietnamese BG_ONLY	eat	meat hedgehog
	"Only the Vietnamese eat hedgehog."
	40.3.3 Givenness
	Like Yongning Na, Vietnamese allows frequent ellipsis of arguments; corpus studies will be
	necessary to quantify its referential density relative to a broad sample of other languages.
	cả		Hàn Quốc cũng	ăn	thịt	chó
	FOC_EVEN		Korean	BG_EVEN/ALSO	eat	meat	dog
	"Even Koreans eat dog meat."

  Chưa thấy bà nội đâu cả. Thằng trộm chạy thoát à? Không, công an bắt rồi.

	(21) thằng	trộm	chạy	thoát	à	không	công an	bắt	rồi
	guy	rob	run	escape	Q	NEG	police	catch already
	"The burglar escaped? No, the police caught him already."
	chưa	thấy	bà		nội	đâu	cả
	not.yet see	grandmother	paternal where	all
	"[I] haven"t seen you (grandmother) anywhere yet". (Hạ 2012:63; excerpt from
	telephone conversations)

Acknowledgments

Many thanks to the Na and Vietnamese consultants and friends, and to the editors and reviewers.

Financial support from Agence Nationale de la Recherche (HimalCo project, ANR-12-CORP-0006, and LabEx EFL, ANR-10-LABX-0083 -Investissements d'Avenir) is gratefully acknowledged.