Mohamed Ali Debyaoui
email: debyaoui@etud.univ-tln.fr

Mehmet Ersoy
email: mehmet.ersoy@univ-tln.fr

A Generalised Serre-Green-Naghdi equations for variable rectangular open channel hydraulics and its finite volume approximation

Keywords: Open channel flow, Euler equations, Asymptotic approximation, Serre-Green-Naghdi equations, Free surface shallow water equations, Non-hydrostatic pressure, Dispersive model, Finite volume

We present a non-linear dispersive shallow water model which enters in the framework of sectionaveraged models. These new equations are derived up to the second order of the shallow water approximation starting from the three-dimensional incompressible and irrotational Euler system. The derivation is carried out in the case of non-uniform rectangular section and it generalises the well-known one-dimensional Serre-Green-Naghdi (SGN) equations on uneven bottom. The section-averaged model is asymptotically consistent with the Euler system in terms of mass, momentum, and energy equation which provides the richness of content for this model. We propose a well-balanced finite volume approximation and we present some numerical results to show the influence of the section variation.

Introduction

In environmental modeling of free surface flows, whenever the aspect-ratio of the domain is small enough, the shallow water approximation is introduced to obtain reduced model for which the computational cost is lower than the one implied by the numerical solution of the full three-dimensional free surface equations. One of the most widely used models to describe the channel and river motion of watercourses is the sectionaveraged free surface model [START_REF] Decoene | Asymptotic derivation of the section-averaged shallow water equations for natural river hydraulics[END_REF][START_REF] Bourdarias | A mathematical model for unsteady mixed flows in closed water pipes[END_REF][START_REF] Ersoy | Dimension reduction for incompressible pipe and open channel flow including friction[END_REF] which is a generalisation of the well-known Saint-Venant system (introduced by Adhémar Jean Claude Barré de Saint-Venant in the 19th Century [START_REF] De Saint-Venant | Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit[END_REF]):

       ∂ t A + ∂ x Q = 0 , ∂ t Q + ∂ x Q 2 A + I 1 (x, A) = I 2 (x, A) . (1)
In these equations, A = σh is the wet area of fluid cross-section, Q is the water discharge,

I 1 (x, A) = A 2 2F 2 r σ
is the hydrostatic pressure where F r is the Froude's number and I 2 (x, A) = σ (x) σ(x)

A 2 2F r 2 σ(x) - A F r 2 d (x)
is the hydrostatic pressure source term which takes into account the variation of the channel width σ and the bottom d. The model [START_REF] Audusse | A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows[END_REF] reduces to the well-known one-dimensional Saint-Venant equations for uniform rectangular section, i.e. if σ is constant. The free surface model is the first order shallow water approximation of the section-averaged Navier-Stokes or Euler equations under suitable assumptions on the horizontal and the vertical scales (see, e.g., [START_REF] Gouta | A finite volume solver for 1d shallow-water equations applied to an actual river[END_REF][START_REF] Decoene | Asymptotic derivation of the section-averaged shallow water equations for natural river hydraulics[END_REF][START_REF] Gerbeau | Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation[END_REF][START_REF] Bourdarias | A mathematical model for unsteady mixed flows in closed water pipes[END_REF][START_REF] Ersoy | Dimension reduction for incompressible pipe and open channel flow including friction[END_REF] and the reference therein).

As it is well-known, the solutions of these equations are usually suitable to approximate breaking waves with turbulent rollers for large transitions of the Froude's number. However, for small or moderate transitions, the solutions of these equations are not able to catch undular bores induced by a non-hydrostatic pressure distribution [START_REF] Peregrine | Calculations of the development of an undular bore[END_REF]. Up to our knowledge, the first section-averaged dispersive shallow water equations for quite general assumptions on the geometry of the channel was proposed in [START_REF] Debyaoui | Generalised Serre-Green-Naghdi equations for open channel and for natural river hydraulics[END_REF], thus allowing for the application of the resulting equations to natural rivers with arbitrarily shaped cross-sections. This model reads

       ∂ t A + ∂ x Q = 0 ∂ t Q + ∂ x Q 2 A + I 1 (x, A) + µ 2 DI 1 (x, A, Q) = I 2 (x, A) + µ 2 DI 2 (x, A, Q) + O(µ 2 2 )
where DI 1 and DI 2 are the non-hydrostatic counterparts of the hydrostatic pressure and the hydrostatic pressure source term. The case of non-uniform rectangular section can be regarded as the natural extension of the usual one-dimensional Serre-Green-Naghdi (SGN) equations over uneven bottom [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF][START_REF] Seabra-Santos | Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle[END_REF][START_REF] Cienfuegos | A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive boussinesq-type equations. part ii: boundary conditions and validation[END_REF].

In this work, we focus only on the case of a rectangular variable section. We first present the geometrical set-up in Sect. 2. Then we give the outline of the asymptotic derivation, and in particular, we show that the section-averaged model is fully consistent with the Euler system in Sect. 3. Finally, in Sect. 4, we construct a first order well-balanced finite volume approximation and we present some numerical test cases.

The three-dimensional Incompressible Euler equations

Settings

We consider the motion of an incompressible and irrotational fluid with constant density ρ 0 > 0 in a three-dimensional domain (see Fig. 1)

Ω(t) = (x, y, z) ∈ R 3 ; x ∈ [0, L c ], α(x) ≤ y -ϕ(x) ≤ β(x), d(x) ≤ z ≤ η(t, x, y)
where ϕ describes the transversal variation of the channel with respect to the main channel direction, α and β are the transversal limits of the channel, L c its length, d is the bottom, η is the free surface and h = ηd is the water height. The boundary of the domain Ω(t) is defined by ∂Ω(t) and is decomposed into four parts: the free surface Γ fs (t), the wet boundary Γ wb (t), the inflow boundary Γ i (t) and the outflow boundary Γ o (t). The wet boundary can be decomposed itself in three parts: the bottom Γ b (t), the left lateral boundary Γ lb (t), and the right one Γ rb (t). The governing equations for the motion of the fluid are the incompressible and irrotational Euler equations in Ω(t), for all t ∈ (0, T], which can be written as follows:

div [u] = 0 , ∂ ∂t (u) + div [u ⊗ u] + ∇ p ρ 0 -F = 0 (2)
where u = (u, v, w) T is the velocity field, F = (0, 0, -g) T is the gravity acceleration and p is the pressure. These equations are completed by the irrotational equation:

curl [u] = 0 . ( 3
)
The system is closed by suitable boundary conditions. We denote by n fs the outward normal to the free surface which depends on time:

n fs = 1 1 + (∂ x η) 2 + ∂ y η 2 -∂ x η, -∂ y η, 1 T ,
and by n wb the outward normal to the wet boundary:

n wb =                          1 1 + (∂ x d) 2 (∂ x d, 0, -1) T if n wb = n b 1 1 + (∂ x α) 2 (∂ x α, -1, 0) T if n wb = n lb 1 1 + (∂ x β) 2 (∂ x β, 1, 0) T if n wb = n rb
At the free surface, we prescribe a kinematic boundary condition

∂ t η + u∂ x η + v∂ y η = w on Γ fs (t) (4)
completed with the dynamical condition which takes into account the equilibrium with atmospheric stress p = p a on Γ fs (t) .

In the sequel, without loss of generality, we set p a = 0. At the wet boundary, we prescribe a no-penetration condition:

u∂ x d -w = 0 on Γ b (t) , u∂ x α -v = 0 on Γ lb (t) , u∂ x β + v = 0 on Γ rb (t) . (6)

Dimensionless Euler equations

Let us consider the following scales involved in the wave motion: L a wave-length in the longitudinal direction, H 2 a characteristic water depth, H 1 a characteristic scale of the channel width and h 1 a wavelength in the transversal direction. We then define the classical dispersive parameter µ 2 (see e.g. [START_REF] Lannes | The Water Waves Problem: Mathematical Analysis and Asymptotics[END_REF])

µ 2 = H 2 2 L 2 and µ 1 = h 2 1
L 2 where µ 1 is also a dispersive parameter but in the transversal direction. In the following, we consider the asymptotic regime:

h 1 < H 1 = H 2 L
such that the following inequality holds µ 1 < µ 2 2 . Under these assumptions, we get the following ordering:

We also introduce This allows us to introduce the dimensionless quantities of time t, space ( x, y, z), pressure P, depth d, water elevation η and velocity field ( u, v, w), via the following scaling relation

(U, V = √ µ 1 U, W = √ µ 2 U) T the
x = x L , y = y h 1 , z = z H 2 , t = t T , P = P P , ϕ = ϕ h 1 , u = u U , d = d H 2 , v = v V , η = η H 2 , w = w W . (7)
Finally, we define the non-dimensional Froude's number by

F r = U √ gH 2 .
For the sake of clarity and simplicity dropping •, using the dimensionless variables [START_REF] Decoene | Asymptotic derivation of the section-averaged shallow water equations for natural river hydraulics[END_REF], and reordering the terms with respect to the powers of µ 1 and µ 2 , the dimensionless incompressible Euler system (2) reads as follows:

∂ x u + ∂ y v + ∂ z w = 0 , (8)
∂ t u + u∂ x u + v∂ y u + w∂ z u + ∂ x P = 0 , (9)
µ 1 ∂ t v + u∂ x v + v∂ y v + w∂ z v + ∂ y P = 0 , (10)
µ 2 ∂ t w + u∂ x w + v∂ y w + w∂ z w + ∂ z P = - 1 F r 2 . (11)
Under this scaling, the boundary conditions ( 4)-( 5) and ( 6) remain unchanged and the dimensionless irrotational equation (3) becomes

∂ y u = µ 1 ∂ x v, µ 1 ∂ z v = µ 2 ∂ y w, ∂ z u = µ 2 ∂ x w . ( 12
)
Thanks to the ordering µ 2 1 < µ 2 and the structure of Eqs. [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF], it is natural to compute the asymptotic expansion of u in two steps first with respect to y, then with respect to z. It can be achieved by first width-averaging the Euler system ( 8)-( 11), then by depth-averaging the resulting equations. For the sake of completeness, skipping the technical details, we present the outline of the derivation. Interested readers can found the details in [START_REF] Debyaoui | Generalised Serre-Green-Naghdi equations for open channel and for natural river hydraulics[END_REF].

Derivation of the section-averaged model

Width-averaged equations

By integrating for s ∈ [α(x), y], the first two equations of the irrotational equations ( 12) and the divergence equation [START_REF] Ersoy | Dimension reduction for incompressible pipe and open channel flow including friction[END_REF], keeping in mind the boundary conditions ( 4)-( 5) and ( 6), we get the following asymptotic expansions:

u(t, x, y, z) = u α (t, x, z) - µ 1 2 ∂ x div x,z w α (t, x, z)(y -α(x)) 2 + O µ 2 1 µ 2 , (13)
v(t, x, y, z) = -div x,z [w α (t, x, z)(y -α(x))] + O µ 1 µ 2 (14)
and

w(t, x, y, z) = w α (t, x, z) - µ 1 2µ 2 ∂ z div x,z w α (t, x, z)(y -α(x)) 2 + O µ 2 1 µ 2 2 ( 15
)
where

X α (t, x, z) := X (t, x, α(x), z).
For a given function (t, x, y, z) → X(t, x, y, z), we define its width-average by

X (t, x, z) := 1 σ(x) ∫ β(x) α(x) X(t, x, y, z) dy where σ(x) = β(x) -α(x) is the width of the channel.
Integrating Eqs. ( 8)-( 11) for y ∈ [α(x), β(x)], using Leibniz integral rule, keeping in mind the boundary conditions ( 4)-( 5) and ( 6), using the asymptotic expansions ( 13)- [START_REF] Lannes | A new class of fully nonlinear and weakly dispersive green-naghdi models for efficient 2d simulations[END_REF], we obtain the width-averaged Euler system:

div x,z [σw α ] = O µ 1 µ 2 , ∂ ∂t (σu α ) + div x,z [σu α w α ] + ∂ ∂ x (σP α )+ = P α ∂σ ∂ x + O µ 1 µ 2 , µ 2 ∂ ∂t (σw α ) + div x,z [σw α w α ] + ∂ ∂z (σP α ) = - σ F r 2 + P α ∂σ ∂z + O(µ 1 ) (16)
where P α (t, x, z)+O(µ 1 ) = P(t, x, y, z) thanks to Eq. [START_REF] Gerbeau | Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation[END_REF]. The motion of the fluid is now in a two-dimensional domain:

Ω (t) = {(x, z) ∈ R; d(x) ≤ z ≤ η * (t, x)} .
The irrotational condition (12) reduces to

∂u α ∂z = µ 2 ∂w α ∂ x + O(µ 1 ) (17)
and the boundary conditions to

∂η * ∂t + u α ∂η * ∂ x = w α + O µ 1 µ 2 and P α = O(µ 1 ) on Γ fs (t) , (18)
u α ∂ x d = w α + O µ 1 µ 2 on Γ b (t) (19)
where Γ fs (t) is the free surface boundary and Γ b (t) the bottom boundary of the width-averaged fluid domain Ω (t).

The function η * in the above expression depends only on t and x. Indeed, integrating Eq. ( 11) for s ∈ [z, η(t, x, y)], using the previous asymptotic expansions, and noting D Dt w = ∂ t w + u∂ x w + v∂ y w + w∂ z w, we can write

P α (t, x, z) = η(t, x, y) -z F r 2 + µ 2 ∫ η(t, x,y) z D Dt w α (t, x, s) ds + O(µ 1 ) .
Thus, taking the y-derivative of the above expression provides

0 = ∂ y η 1 F 2 r + µ 2 D Dt w α |z=η + O(µ 1 ) = -∂ y η ∂ z P |z=η + O(µ 1 )
Consequently, since ∂ z P |z=η 0, we get ∂ y η = O(µ 1 ). This is the so-called flat free surface approximation. Therefore, one can write

η(t, x, y) = η * (t, x) + O(µ 1 ) (20)
where the * is dropped in the following.

Depth-averaged equations

Integrating Eq. ( 17) together with the first equation of System ( 16) for s ∈ [d(x), z], keeping in mind Eqs. ( 18)-( 19), we obtain

u α (t, x, z) = u d (t, x) -µ 2 ∫ z d(x) ∂ x S(u d , x, s) ds + O(µ 2 2 )
and

w α (t, x, z) = - 1 σ(x) ∂ ∂ x (u d (t, x)S(x, z)) + O(µ 2 )
where S(u,

x, z) = 1 σ(x) ∂ ∂ x (uS(x, z)), S(x, z) = σ(x)(z -d(x)) and X d (t, x) = X α (t, x, d(x)).
Thanks to the flat free surface approximation (20), one can write the section-average of the velocity u as follows:

u = 1 A ∫ η(t, x) d(x) ∫ β(x) α(x) u(t, x, y, z) dy dz where A = ∫ η(t, x) d(x) ∫ β(x) α(x) dy dz = σ(x)h(t, x) is the wet area, σ = β -α is the width of the channel and h = η -d is the water height. Thus, since u(t, x, y, z) = u α (t, x, z) + O(µ 1 ) = u d (t, x) -µ 2 ∫ z d(x) ∂ x S(u d , x, s) ds + O(µ 2
2 ), we deduce the following asymptotic expansion of u:

u = u(t, x) + µ 2 B 0 (u, x, z) + O(µ 2 2 ) (21)
where

B 0 (u, x, z) = 1 A(t, x) ∫ η(t, x) d(x) σ(x) ∫ z d(x) ∂ x S(u, x, s) ds dz - ∫ z d(x) ∂ x S(u, x, s) ds .
Similarly, we get for w:

w(t, x, y, z) = -S(u, x, z) + O µ 1 µ 2 . ( 22
)
Using the asymptotic expansion of u (21) and w (22) , we obtain the asymptotic expansion of the pressure P at order O(µ 2 2 )

P(t, x, y, z) = P α (t, x, z) + O(µ 1 ) = P h (t, x, z) + µ 2 P nh (t, x, z) + O(µ 2 2 )
where

P h (t, x, z) = (η(t, x) -z) F r 2
is the usual hydrostatic pressure and

P nh (t, x, z) = ∫ η(t, x) z 1 2σ(x) 2 ∂ z (σ(x)S(u, x, s)) 2 ds - ∫ η(t,x) z ∂ t S(u, x, s) + u(t,x) σ(x) ∂ x (σ(x)S(u, x, s)) ds
is the non-hydrostatic part of the pressure.

Section-averaged model

To end the asymptotic derivation, we integrate vertically the set of equations ( 16) between d and η and drop all terms of order lower than µ 2 . We get the generalised Serre-Green-Naghdi equations for non-uniform rectangular section:

       ∂ t A + ∂ x Q = 0 ∂ t Q + ∂ x Q 2 A + I 1 (x, A) + µ 2 DI 1 (x, A, Q) = I 2 (x, A) + µ 2 DI 2 + O(µ 2 2 ) (23)
where A = σh is the wet area, Q is the water discharge, 8)- [START_REF] Gouta | A finite volume solver for 1d shallow-water equations applied to an actual river[END_REF]. We have the following result: Theorem 1 System (23) admits a total energy

I 1 (x, A) = A 2 2F 2 r σ(x) is the hydrostatic pressure, I 2 (x, A) = σ (x) σ(x) A 2 2F r 2 σ(x) - A F r 2 d (x)
E = A u 2 2 + A η F 2 r -I 1 + µ 2 2 ∫ Ω S 2 (u, x, z) dydz ( 24
)
which satisfies the following energy equation

∂ t E + ∂ x ((E + I 1 + µ 2 DI 1 )u) = 0 . ( 25
)
Moreover, the quantity E is consistent with the total energy

E = u 2 +µ 1 v 2 +µ 2 w 2 2 + z F 2 r
of the Euler equation (8)- [START_REF] Gouta | A finite volume solver for 1d shallow-water equations applied to an actual river[END_REF], in the sense that

∂ t ∫ Ω E dydz + ∂ x ∫ Ω (E + P)u dydz = ∂ t E + ∂ x ((E + I 1 + µ 2 DI 1 )u) + O(µ 2 2 )
.

Remark 1 This is a positive feature of the approximate model ( 23), which provides the richness of content for this model and can be used in the estimation of the accuracy of numerical algorithms. Moreover, it is well-known that the energy conservation law plays a fundamental role in the justification of the theory of shallow water equations.

Remark 2 As a direct consequence of ( 24) and (25), we are able to recover the energy conservation law of the usual models in the case of σ ≡ 1, i.e. A = h:

• if µ 2 = 0, we recover the classical total energy of the Saint-Venant system, namely

E = hu 2 2 + h(h + 2d) 2F 2 r .
• if µ 2 0, we recover the classical total energy of the Serre-Green-Naghdi system (see for instance [START_REF] Fedotova | Energy equation for certain approximate models of long-wave hydrodynamics[END_REF]), namely

E = hu 2 2 + h(h + 2d) 2F 2 r + µ 2 h 3 6 (∂ x u) 2 -d h 2 2 ∂ x u + (d ) 2 2 .

A well-balanced finite volume approximation

The main drawback of Eqs. ( 23) is that it has third order terms in space which may lead to instabilities at the numerical level. Therefore, we first propose a more stable formulation of Eqs. (23) before presenting its numerical approximation. Skipping the technical details, defining a linear operator L (where L is defined below)

L[A, d, σ](u) = AL[A, d, σ] u A ,
one can show that System (23) can be written:

∂ t A + ∂ x Q = 0 I d -µ 2 L[A, d, σ] ∂ t (Au) + ∂ x Q 2 A + ∂ x I 1 (x, A) + µ 2 AQ[A, d, σ] Q A = I 2 (x, A) + O(µ 2 2 ) (26)
where Q = Au is the discharge, I d is the identity operator, L is a linear operator

L[A, d, σ](u) = 1 A ∂ x T [A, d, σ] (u, σ) -T [A, d, σ] (u, ∂ x σ) + 1 A σ(x)d (x) T [A, d, σ, z = d(x)] (u)
and Q is a quadratic operator

Q[A, d, σ](u) = 1 A ∂ x G[A, d, σ] (u, σ) -G[A, d, σ] (u, ∂ x σ) + 1 A σ(x)d (x) G[A, d, σ, z = d(x)] (u)
with T , G are given by

T [A, d, σ, z](u) = ∂ x (u) ∫ η z S(x, s) σ(x) ds + u ∫ η z 1 σ(x) ∂ x S(x, s) ds , and
G[A, d, σ, z](u) = ∫ η z 2 (∂ x u) 2 S(x,s) σ(x) + u 2 σ(x) ∂ x S(x,s)∂ x σ(x) σ(x) -∂ x ∂ x S(x, s) + ∂ x u 2 2 S(x,s)∂ x σ(x) σ(x) 2 ds with X[A, d, σ](u, ψ) = ∫ η d(x) ψX[A, d, σ, z](u) dz .
In particular, one can explicitly compute those operators:

• if σ ∈ R + * and d ∈ R are constant then we recover the standard one-dimensional SGN equations (see for instance [START_REF] Lannes | Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation[END_REF][START_REF] Lannes | A new class of fully nonlinear and weakly dispersive green-naghdi models for efficient 2d simulations[END_REF][START_REF] Lannes | Nonlinear wave-current interactions in shallow water[END_REF]) over flat bottom with

L[A, d, σ](u) = L 0 [A, σ](u) = 1 σh ∂ x σh 3 3 ∂ x u and Q[A, d, σ](u) = Q 0 [A, σ](u) = 1 σh ∂ x 2 3 σh 3 (∂ x u) 2 .
• if σ ∈ R + * is constant and d = d(x) then we recover the standard one-dimensional SGN equations (see for instance [START_REF] Lannes | Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation[END_REF][START_REF] Lannes | A new class of fully nonlinear and weakly dispersive green-naghdi models for efficient 2d simulations[END_REF][START_REF] Lannes | Nonlinear wave-current interactions in shallow water[END_REF]) over uneven bottom with

L[A, d, σ](u) = L 1 [A, d, σ](u) = L 0 [A, σ](u) - 1 σh ∂ x σh 2 2 ud (x) + h 2 ∂ x ud (x) -u (d (x)) 2
and

Q[A, d, σ](u) = Q 1 [A, d](u) = Q 0 [A, σ](u) + 1 σh ∂ x σ h 2 2 u 2 d (x) + h (∂ x u) 2 d (x) + u 2 d (x) d (x) .
• if σ = σ(x) and d = d(x) then we get the generalised one-dimensional SGN equations for non-uniform rectangular channel over uneven bottom with

L[A, d, σ](u) = L 1 [A, d, σ](u) + 1 σh ∂ x σ (x) h 3 3 u - σ (x) σ ∂ x u h 2 3 + u h 2 3 σ (x) σ -u h 2 d (x) and Q[A, d, σ](u) = Q 1 [A, d, σ](u) + 1 σh ∂ x (σ (x)) 2 u 2 σ h 3 3 + 1 σh ∂ x d (x)σ (x)u 2 h 2 2 -1 σh ∂ x σ (x)u 2 h 3 3 + ∂ x ∂ x u 2 2 σ (x) h 3 3 -1 σh σ (x)R[A, d, σ](u) with R[A, d, σ](u) = (∂ x u) 2 h 3 3 + u 2 σ (x) σ 2 h 3 3 + u 2 σ (x) σ d (x) h 2 2 -u 2 σ (x) σ 2 h 3 3 + u 2 d (x) h 2 2 +∂ x u 2 2 σ (x) σ h 3 3 -u 2 d (x) σ (x) σ h 2 2 -u 2 σ (x) (d (x)) 2 h + u 2 σ (x)d (x) h 2 2 -∂ x u 2 2 σ (x)d (x) h 2 2 .
It is known that third order derivatives involved in the initial model ( 23) may create high frequencies instabilities, but the presence of the

I d -µ 2 L[A, d, σ] -1
in the second equation of (26) stabilises the equations with respect to these perturbations. Therefore, in the following, we construct a numerical scheme for Eqs. (26) instead of Eqs. (23).

Numerical method

This section is devoted to the numerical method to solve the reformulated dispersive model (26). It is rather natural to split the hyperbolic part to the dispersive one as done by several authors (see for instance [START_REF] Cienfuegos | A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive boussinesq-type equations. part ii: boundary conditions and validation[END_REF][START_REF] Chazel | Numerical simulation of strongly nonlinear and dispersive waves using a green-naghdi model[END_REF][START_REF] Bourdarias | A numerical scheme for an improved green-naghdi model in the camassa-holm regime for the propagation of internal waves[END_REF]).

Let N ∈ N * . Let us consider the following uniform mesh on [0, L c ]. Cells are denoted for every

i ∈ [0, N + 1], by m i = (x i-1/2 , x i+1/2 ) with x i = x i-1/2 +x i+1/2 2
the cell center and δx = x i+1/2x i-1/2 the space mesh. The interfaces x 1/2 = 0 and x = x N +1/2 denote the upstream and the downstream ends. We also consider a time discretisation t n defined by t n+1 = t n + δt n where the time step δt n is computed through a CFL condition related to the hyperbolic part.

Let us first highlight that the still water steady state for Eqs. ( 26) is independent of µ 2 . Indeed, one has ∀µ 2 > 0, the still water steady state equation reads

u = 0, A σ + d = h 0
for some positive h 0 . As a consequence, the construction of a well-balanced scheme can be easily achieved considering only the hyperbolic part of Eqs. ( 26), for instance, by the use of the hydrostatic reconstruction (see for instance [START_REF] Audusse | A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows[END_REF]).

Let us define

d i+1/2 = max(d i , d i+1 ) where d i = 1 δx ∫ m i d(x)dx, σ i+1/2 = max(σ i , σ i+1 ) where σ i = 1 δx ∫ m i σ(x)
dx and let us define the reconstructed states

A - i+1/2 = σ i+1/2 A i σ i + d i -d i+1/2 , A + i+1/2 = σ i+1/2 A i+1 σ i+1 + d i+1 -d i+1/2 with U - i+1/2 = (A - i+1/2 , A - i+1/2 u i ), U + i+1/2 = (A + i+1/2 , A + i+1/2 u i+1 ) where U i = (A i , A i u i ) T ≈ 1 δx ∫ m i (A, Au) T dx. Let us introduce the flux F 1 (U) = Q, F 2 (U) = Q 2 /A and F 3 (x, U) = I 1 (x, A) + µ 2 G[A, d, σ] (u, σ) and S(x, U) = I 2 + µ 2 G[A, d, σ] (u, ∂ x σ) -µ 2 σ(x)d (x) G[A, d, σ, z = d(x)] (u) .
Then, one can write System (26) as follows:

∂ t A + ∂ x F 1 (U) = 0 I d -µ 2 L[A, d, σ] (∂ t Q + ∂ x F 2 (U)) + ∂ x F 3 (x, U) -S(x, U) = 0
With these settings, we define the following numerical scheme:

A n+1 i = A n i -δt n δx F 1 U -,n i+1/2 , U +,n i+1/2 -F 1 U -,n i-1/2 , U +,n i-1/2 Q * i = Q n i -δt n δx F 2 U -,n i+1/2 , U +,n i+1/2 -F 2 U -,n i-1/2 , U +,n i-1/2 Q n+1 i = Q * i -δt n δx (Y n ) i where A n Y n = F 3 x i+1/2 , U -,n i+1/2 , U +,n i+1/2 -F 3 x i-1/2 , U -,n i-1/2 , U +,n i-1/2 + µ 2 N n i 1≤i ≤ N .
The matrix A n is the cell-centered approximation of the linear operator

I d -µ 2 L[A, d, σ] and N n i is the cell-centered approximation of -G[A, d, σ] (u, ∂ x σ) + σ(x)d (x) G[A, d, σ, z = d(x)] (u).
The numerical fluxes are defined by

F 1 U -,n i+1/2 , U +,n i+1/2 = F 1 U -, n i+1/2 +F 1 U +, n i+1/2 2 -s n i+1/2 (A +,n i+1/2 -A -,n i+1/2 ) F 2 U -,n i+1/2 , U +,n i+1/2 = F 2 U -, n i+1/2 +F 2 U +, n i+1/2 2 -s n i+1/2 (Q +,n i+1/2 -Q -,n i+1/2 ) F 3 x i+1/2 , U -,n i+1/2 , U +,n i+1/2 = F 3 x i+1/2 ,U -, n i+1/2 +F 3 x i+1/2 ,U +, n i+1/2 2 + A n i 2 2σ i F 2 r - A -, n i+1/2 2 2σ i+1/2 F 2 r F 3 x i+1/2 , U -,n i+1/2 , U +,n i+1/2 = F 3 x i+1/2 ,U -, n i+1/2 +F 3 x i+1/2 ,U +, n i+1/2 2 + A n i+1 2 2σ i+1 F 2 r - A +, n i+1/2 2 2σ i+1/2 F 2 r
such that whenever µ 2 = 0, we recover the classical numerical scheme1 for the hyperbolic part

U n+1 i = U n i - δt n δx F (x i+1/2 , U -,n i+1/2 , U +,n i+1/2 ) -F (x i-1/2 , U -,n i-1/2 , U +,n i-1/2 ) with F (x, U, V) = (F 1 (U, V), F 2 (U, V) + F 3 (x, U, V)).
In these expressions,

s i+1/2 = max j=1,2 λ j (x i+1/2 , U -,n i+1/2 ) , λ j (x i+1/2 , U +,n i+1/2 )
where

λ j (x, U) = Q/A + (-1) j A σ(x)F 2 r
, j = 1, 2 are the eigenvalues of the Jacobian matrix of (F 1 , F 2 + F 3 ) T . The numerical scheme is consistent and stable under the CFL condition

max 1≤i ≤ N λ 1 (x i , U n i ) , λ 2 (x i , U n i ) δt n δx ≤ 1 .

Propagation of a solitary wave

In this section, we test the accuracy of the method and we show numerically the influence of the section variation in the case of the propagation of a solitary wave. For this purpose, we consider the exact solitary wave solutions of the Green-Naghdi equations in the one-dimensional setting over a flat bottom (see [START_REF] Lannes | A new class of fully nonlinear and weakly dispersive green-naghdi models for efficient 2d simulations[END_REF]), given in variables with dimensions, by

η(t, x) = asech 2 (k(x -ct)), u(t, x) = c η(t, x) η(t, x) + z 0 with k = √ 3a 2z 0 √ z 0 + a and c = g(z 0 + a) (27)
where z 0 is the depth of the fluid and a is the relative amplitude.

Accuracy

The propagation of the solitary wave ( 27) is initially centered at x 0 = 10 m with a relative amplitude a = 0.2 m over a constant water depth z 0 = 2 m. The computational domain is L c = 100 m and it is discretized with N cells. The single solitary wave propagates from left to right. In this test, since the solitary wave is initially far from boundaries, the boundary conditions do not affect the computation, thus we choose to impose free boundary conditions at the downstream and upstream ends. The exact solution is computed in a channel of width σ = 1.

In what follows, we quantify the numerical accuracy of our numerical scheme by computing the numerical solution for this particular test case for an increasing number of cells N over a duration T = 20 s. Starting with N = 100 number of cells, we successively multiply the number of cells by two. For all n, we compare, in Fig. 2, M n := max 0≤i ≤N +2 (h n i ) of our numerical solution provided by Eqs. ( 26) with the exact one M(t n ) := max h(t n , x) x ∈[0, L c ] = 2.2 given by (27). One can easily remark that the first order discretisation is not accurate for long time simulation due to the numerical dissipation. However, to limit the numerical dissipation of the first order numerical scheme, one can either limit the simulation time or consider a very large number of cells. However, it is better to increase the order of the numerical scheme but this is left to future work. Therefore, in what follows, we consider a shorter simulation time and a large number of cells, just to illustrate the influence of the variation of the channel. for each geometry, we show the evolution of the maximum of the water level M n := max 0≤i ≤N +2 (h n i ). As expected, since the first part for x ≤ 25 is linearly converging, the water level increases while for x > 25, the channel is linearly diverging and therefore, the amplitude of the water level decreases. Moreover, in all numerical simulations, the mass is conserved. Indeed, for each value of ε, we have displayed in Fig. 4(b), the ratio of m n m 0 where m

n := 1 N +2 N +1
i=0 A n i is the mass of water at time t n . The ratio m n m 0 is almost equal to 1, up to the order of accuracy of the numerical scheme.

ε = 0 ε = 0.1 ε = 0.2 ε = 0.3 ε = 0.4 (a) M n := max x ∈[0, Lc ] (h n i ) 0
ε = 0 ε = 0.1 ε = 0.2 ε = 0.3 ε = 0.4 (b) m n m 0 with m n = 1 N +2 N +1 i=0 A n i
In what follows, we quantify the numerical accuracy of our numerical scheme. Starting with N = 100 number of cells, we successively multiply the number of cells by two. The errors on the water surface deformation are presented in Table 1 and in Table 2. These errors are computed at t = 8 s using the L 2 :

η num -η ref 2 = δx i η numi (t = 8) -η ref (t = 8, x i )
and the L ∞ norms where η ref is the exact solution in the case ε = 0 and is a reference one computed with 10 000 cells for ε = 0.4. Since the results are almost the same whatever ε is, we have decided to present only the results for ε = 0 and ε = 0.4 in Table 1 and in Table 2.

As expected, the obtain numerical order is slow because of the numerical dissipation of the solitary wave (as already pointed out in several works, see for instance [START_REF] Bourdarias | A numerical scheme for an improved green-naghdi model in the camassa-holm regime for the propagation of internal waves[END_REF] for which we obtain almost the same order of convergence in the case of uniform section). Moreover, as expected, cross-sectional variations have no influence on the convergence rate. Let us just emphasise that the convergence rates are slightly better in the case of non-uniform section because we are comparing our results to a reference solution and not to the exact one.

Conclusions and perspectives

We have presented the derivation of a new dispersive model for open channel with non-uniform rectangular section. This model generalises the usual Serre-Green-Naghdi equation. We have presented its numerical finite volume approximation for which we have proposed two simple test cases. In a forthcoming paper, we will focus on the case of arbitrary channel section and we will propose a high order numerical scheme.

Fig. 1

 1 Fig. 1 Geometric set-up

  scale of fluid velocity. The time scale is T = L U . Let us define P = p ρ 0 and choose the pressure scale to be P = U 2 .

Fig.

  Fig.2M n := max 0≤i≤ N +2 (h n i )

2 - ε 2 exp -ε 2 x

 222 We consider again the propagation of a solitary wave initially centered at x 0 = 10 m of relative amplitude a = 0.2 m, over a constant water depth z 0 = 2 m onto a computational domain of L c = 50 m and discretized with N = 5000 cells. The final simulation time is T = 8 s. Initially starting with (η(0, x), u(0, x)) (see Eqs. (27)), we compute the numerical simulation for the channels defined by σ(x; ε) = β(x; ε) -α(x; ε) with β = 1 -L/2) 2 and α = -β with = 0, = 0.1, = 0.2, = 0.3 and = 0.4. The obtained results are presented in Fig. 3. In Fig.4(a),

Fig.

  Fig. 3 Influence of σ

  2 M n := max 0≤i≤ N +2 (h n i )

	2.2
	2.19
	2.18
	2.17
	2.16
	2.15
	2.14
	2.13	0	2	4	6	8	10	12	14	16	18	20

3

  Influence of σ

		2.215
		2.21
		2.205
		2.2
	M n	2.195
		2.19
		2.185
		2.18
		2.175	0	1	2	3	4	5	6	7	8
							t (s)

Table 1

 1 Convergence rate of the L 2 error for ε = 0. The order is computed through first order interpolation polynomial N η num -η e x a c t 2 η num -η e x a c t ∞

	100	0.0789	0.0449
	200	0.0497	0.0288
	400	0.0304	0.0180
	800	0.0198	0.0116
	1600	0.0153	0.0081
	3200	0.0138	0.0062
	Order	0.53	0.58

Table 2

 2 Convergence rate of the L 2 error for ε = 0.4. The reference solution is computed with 10 000 cells. The order is computed through first order interpolation polynomialN η num -η ref 2 η num -η ref ∞

	100	0.05212	0.02533
	200	0.02096	0.01082
	400	0.01079	0.00554
	800	0.00748	0.00503
	1600	0.00635	0.00412
	3200	0.00505	0.00300
	Order	0.64	0.56

For the sake of simplicity and clarity, we have presented the finite volume method using the Rusanov solver but the method is not limited to this one.

Acknowledgment

The authors thank the referees for their valuable remarks which led to a substantial improvement of the first version of this paper. The second author wishes to grateful Dr Griggio for her help throughout this work and would like to say sincerely "Sağol".


Lian Lian

Rosa Del

C Ortiz

Florian Jabbour

Zhi-Duan Chen

Wei Wang

Re-delimitation of Tinospora (Menispermaceae): Implications for character evolution and historical biogeography

Keywords: biogeography, Fawcettia, geodispersal, Menispermaceae, phylogeny, tropical forests

Traditional Tinospora is a heterogeneous genus, distributed in the African and Asian tropics as well as Australia and adjacent islands. Molecular phylogenetics has contributed greatly to improve our understanding of the genus over the past decade, however its circumscription is yet unresolved. In this study, we present a phylogenetic analysis of Burasaieae using six molecular markers (rbcL, atpB, matK, ndhF, trnL-F, ITS), with a focus on Tinospora. Our results indicate that sampled species of Tinospora are distributed in three different clades. Tinospora tinosporoides, endemic to Australasia, is strongly supported as sister to the South American Borismene. By integrating lines of evidence frommolecular phylogeny andmorphology, we resurrected themonotypic Fawcettia to include F. tinosporoides. Our estimated age for the split of Fawcettia-Borismene was ca. 29 Ma, suggesting that trans-Pacific long-distance dispersal may be the most feasible explanation for the Australasian-South American disjunction. Tinospora oblongifolia is a member of Hyalosepalum, characterized by three stamens with connate filaments, whereas T. tenera, once placed in Hyalosepalum, is recovered within Tinospora, characterized by six stamens. Evolutionary inferences of morphological characters indicate that the five selected characters, traditionally used to distinguish genera in Burasaieae, are highly homoplastic. Our molecular dating and ancestral range reconstruction suggest that Tinospora originated from Africa in the middle Oligocene (ca. 28 Ma) and expanded into Asia in the Early Miocene (ca. 21 Ma). Within Tinospora, a vicariance event occurred in the Middle Miocene (ca. 17 Ma) resulting in the split of African T. tenera and its Asian allies. One dispersal fromAsia toAfricawas inferred in the LateMiocene (ca. 10 Ma). Australian T. smilacina and T. esiangkara originated from Asia in the early Late Miocene (ca. 11 Ma). Thus, geodispersal and subsequent vicariance, as well as transoceanic long-distance dispersal, have been responsible for the present distribution of Tinospora.

INTRODUCTION

The genus Tinospora Miers (Menispermaceae), consisting of about 32 species, is disjunctly distributed in the African and Asian tropics, also in Australia and adjacent islands [START_REF] Forman | Menispermaceae[END_REF][START_REF] Forman | Flora of Australia[END_REF][START_REF] Kessler | Menispermaceae[END_REF]. Tinospora species are main foodplants of larvae of fruit-piercing moths, and thus play an important role in the biological chain that results in extensive damage to certain fruit crops, such as longan, citrus, and mandarin [START_REF] Bänziger | Fruit-piercing moths (Lep., Noctuidae) in Thailand: A general survey and some new perspectives[END_REF]. Some species, such as T. crispa (L.) Hook.f. & Thomson, T. sinensis (Lour.) Merr., and T. sagittata Gagnep. (= Paratinospora sagittata (Oliv.) Wei Wang), are utilized as traditional Chinese medicines [START_REF] Luo | Menispermaceae[END_REF]. Nevertheless, traditional Tinospora is a heterogeneous assemblage. Based on molecular phylogenetic analyses, at least four species have been segregated from the genus. Tinospora caffra (Miers) Troupin (= Desmonema caffra Miers) was the first to be found not to cluster with other Tinospora [START_REF] Ahmad | Phylogenetic patterns and genetic diversity of Indian Tinospora species based on chloroplast sequence data and cytochrome P450 polymorphisms[END_REF]Hoot & al., 2009;[START_REF] Jacques | Integrating fossils in a molecular-based phylogeny and testing them as calibration points for divergence time estimates in Menispermaceae[END_REF][START_REF] Wefferling | Phylogeny and fruit evolution in Menispermaceae[END_REF][START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF]. [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF] confirmed this result and resurrected Hyalosepalum Troupin to accommodate H. caffrum (Miers) Troupin (= T. caffra). Tinospora sagittata Gagnep. (= Paratinospora sagittata (Oliv.) Wei Wang) and T. dentata Diels (= Paratinospora dentata (Diels) Wei Wang) were excluded from Tinospora and placed into the newly described genus Paratinospora Wei Wang [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF]. Tinospora uviforme (Baill.) Troupin was recovered as a member of Chasmanthera Hochst., i.e., C. uviformis Baill. [START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF]. On the other hand, Limacia esiangkara F.M.Bailey was transferred to Tinospora by [START_REF] Forman | Flora of Australia[END_REF] as T. esiangkara (F.M.Bailey) Forman, and this placement was confirmed by subsequent molecular studies [START_REF] Ortiz | Molecular phylogeny of the moonseed family (Menispermaceae): Implications for morphological diversification[END_REF][START_REF] Wang | Menispermaceae and the diversification of tropical rainforests near the Cretaceous-Paleogene boundary[END_REF][START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF]. Although tremendous progress has been made in our understanding of Tinospora over the past ten years, its circumscription has remained unresolved, mainly owing to the uncertain systematic position and taxonomic treatment of Fawcettia F.Muell.

Fawcettia with the sole species F. tinosporoides F.Muell. was first described by Mueller (1877) and was thought to be close to Fibraureae or Tinospora of Burasaieae. The genus was recognized by [START_REF] Bailey | The Queensland flora, part 1[END_REF] and was retained by [START_REF] Diels | Menispermaceae[END_REF] in his monograph of the family Menispermaceae, on the basis of the presence of a third whorl of outermost sepals. However, [START_REF] Forman | A revision of Tinospora (Menispermaceae) in Asia to Australia and the Pacific[END_REF] argued that the presence of this third outermost whorl of sepals was not only inconsistent in Fawcettia, but that the difference in sepal whorls was too small to justify the recognition of the latter and therefore placed it in Tinospora as T. tinosporoides (F.Muell) Forman. Subsequent authors accepted its inclusion in Tinospora [START_REF] Kessler | Menispermaceae[END_REF][START_REF] Forman | Flora of Australia[END_REF]Mabberley, 2017). Tinospora tinosporoides is restricted to rainforests and complex notophyll vine forests in southeastern Queensland and northeastern New South Wales [START_REF] Forman | Flora of Australia[END_REF], and to date, this species has not been sampled in any molecular study.

Tinospora together with Chasmanthera Hochst. and Jateorhiza Miers was placed in a broad Chasmanthera by [START_REF] Baillon | [END_REF]. However, this broad Chasmanthera was not accepted by subsequent authors (e.g., [START_REF] Diels | Menispermaceae[END_REF][START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF][START_REF] Kessler | Menispermaceae[END_REF]. [START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF] argued, if Baillon' s concept was adopted, Kolobopetalum Engl., Leptoterantha Louis ex Troupin, and Syntriandrium Engl. should also be reduced to the broad Chasmanthera. Similarly, Odontocarya Miers could also be placed in the broad Chasmanthera [START_REF] Barneby | Revision of Neotropical Menispermaceae tribe Tinosporeae[END_REF]. Sepals and stamens are considered the most important characters in keeping these genera separate [START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF]), yet to date, their taxonomic significance has never been discussed in comparison to molecular phylogeny. Recent phylogenetic analyses indicate that the aforementioned seven genera, as well as Dialytheca Exell&Mendonça, Hyalosepalum, Rhigiocarya Miers, and Sarcolophium Troupin, formed a monophyletic group with strong support (e.g., [START_REF] Wang | Menispermaceae and the diversification of tropical rainforests near the Cretaceous-Paleogene boundary[END_REF][START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF][START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF]. To shed light on the taxonomic status of these genera, the evolutionary patterns of some taxonomically relevant morphological characters, such as number of sepal whorls, stamen connation, and pollen aperture types [START_REF] Diels | Menispermaceae[END_REF][START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF][START_REF] Kessler | Menispermaceae[END_REF], need to be evaluated in a broad phylogenetic context.

The objectives of this study are: [START_REF]Taxa, voucher identification and GenBank accession numbers for molecular analyses of the tribe Burasaieae. Species, locality, voucher, and GenBank accession: ITS, trnL-F, matK, ndhF, rbcL, and atpB. An asterisk (*) indicates newly generated sequences in this study, a dash (-) denotes missing data[END_REF] to re-delimitate Tinospora by clarifying the phylogenetic position and taxonomic treatment of Fawcettia, (2) to determine the evolutionary patterns of key morphological characters of Burasaieae, and (3) to investigate the historical biogeography of Tinospora s.str.

MATERIALS AND METHODS

Taxon sampling and DNA sequencing

We sampled 44 species, representing 22 of the 24 currently recognized genera in Burasaieae [START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF], and Paratinospora and Hyalosepalum recently described and resurrected by [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF], respectively. Our sampling includes 16 species still placed in Tinospora, 3 from Africa, 4 from Australasia, and 9 from Asia. Following the results of [START_REF] Wang | Menispermaceae and the diversification of tropical rainforests near the Cretaceous-Paleogene boundary[END_REF] and [START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF], we selected three species representing all three genera of Coscinieae, the other tribe of Chasmantheroideae, as outgroups. Species, vouchers and GenBank accession numbers are given in the Appendix 1.

We amplified six plastid (rbcL, atpB, matK, ndhF, trnL-F) and nuclear (ITS) markers. Laboratory procedures and sequence handling followed [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF]. When ITS-A and ITS-B did not work for long ITS sequences, additional primers, ITSC and ITS-D, were used [START_REF] Blattner | Direct amplification of the entire ITS region from poorly preserved plant material using recombinant PCR[END_REF]. Sequences were aligned using Geneious v.10. 1.3 (Kearse & al., 2012) based on the previous data matrices of [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF], and then adjusted manually in BioEdit v.7.0 [START_REF] Hall | BioEdit: A user-friendly biological sequence alignment editor and analysis program for Windows 95/98/NT[END_REF]. One polyAG region in the trnL-F matrix (representing 10 nucleotides) and four difficult-to-align regions in the ITS matrix (encompassing 126 nucleotides) were excluded from the analyses. The final lengths of these six marker datasets were : rbcL, 1385 bp;atpB, 1412 bp;matK, 1235 bp;ndhF, 2066 bp;trnL-F, 1133 bp;and ITS, 656 bp (suppl. Appendices S1-S6).

Phylogenetic analysis

Phylogenetic analyses were conducted using maximum likelihood (ML) and Bayesian inference (BI) methods for the combined cpDNA (five plastid markers), ITS, and the combined cpDNA and ITS datasets. ML and BI analyses were conducted in RAxML v.7.2.6 [START_REF] Stamatakis | RAxML-VI-HPC: Maximum likelihood-based phylogenetic analyses with thousands of taxa and mixed models[END_REF] and MrBayes v.3.2.5 [START_REF] Ronquist | MrBayes 3.2: Efficient Bayesian phylogenetic inference and model choice across a large model space[END_REF], respectively. For ML analyses, RAxML was performed with the GTR + I + Γ substitution model for each DNA region, and nodal support was assessed using the fast bootstrap option with 1000 replicates. For BI analyses, each DNA region was assigned its own model of nucleotide substitution, as determined by the Akaike information criterion (AIC) via jModeltest v.2. 1.4 (Posada, 2008). Two independent runs, each consisting of fourMarkov ChainMonte Carlo (MCMC) chains, were conducted with one tree sampled for every 1000 generations over 50 million generations, starting with a random tree. Stationarity was determined in Tracer v. 1.6 (Rambaut & al., 2014). Majority-rule (>50%) consensus trees were constructed after removing the burn-in period samples (the first 25% of sampled trees). Based on [START_REF] Wang | The analysis of chloroplast and nuclear DNA data in plant phylogenetics-Evaluation and improvement[END_REF], we used the thresholds BS ≥70%and PP ≥0.95 as an indication of significant conflict between the cpDNA and ITS datasets.

Analysis of character evolution

The distribution of five morphological characters, upon which the generic classification of Burasaieae, especially the broad Chasmanthera has traditionally been based, was investigated. Orthogynium sp. and Tinospora sp. were excluded from the analyses because the five characters evaluated here are still unknown in these two taxa. Information on morphological features was taken from the literature, particularly [START_REF] Diels | Menispermaceae[END_REF], [START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF], [START_REF] Barneby | Revision of Neotropical Menispermaceae tribe Tinosporeae[END_REF], [START_REF] Forman | A revision of Tinospora (Menispermaceae) in Asia to Australia and the Pacific[END_REF], and [START_REF] Kessler | Menispermaceae[END_REF] for general morphological characters, and [START_REF] Thanikaimoni | Menispermacées: Palynologie et systématique[END_REF] for pollen aperture types. Patterns of evolution were inferred using the ML approach as implemented in Mesquite v.2.75 [START_REF] Maddison | Mesquite, version 2.75[END_REF]. The Markov k-state one-parameter model of evolution for discrete unordered characters [START_REF] Lewis | A likelihood approach to estimating phylogeny from discrete morphological character[END_REF] was used.

Molecular dating analysis

We estimated the divergence times of Burasaieae using the uncorrelated lognormal Bayesian method in BEAST v.2. 1.3 (Drummond & al., 2012). Following [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF], we used the cpDNA data for age estimates owing to the potential problems of recombination, incomplete concerted evolution, and high homoplasy in ITS sequences. Phylogenetic analyses [START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF][START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF]this study) indicate that the species of Tinospora are distributed in at least five clades. Thus, we did not use T. excavata Reid & Chandler from the Paleocene-Eocene transition of eastern Belgium [START_REF] Fairon-Demaret | Fruits and seeds from the Tienen Formation at Dormaal, Paleocene-Eocene transition in eastern Belgium[END_REF] as a calibration point, as it is not possible to place it in the phylogenetic tree. Herewe selected three fossils that can confidently be placed in our tree as internal calibration points. (1) A fossil endocarp of Anamirta Colebr. was used to constrain its stem age to at least 43.7 Ma [START_REF] Manchester | Fruits and seeds of Middle Eocene Nut Beds Flora, Clarno formation Oregon[END_REF]. ( 2) The stem age of Parabaena Miers was constrained to be at least 47.8 Ma based on the fossil Parabaena bognorensis Chandler from the London Clay [START_REF] Chandler | The Lower Tertiary Floras of southern England IV: A summary and survey of findings in the light of recent botanical observations[END_REF]. (3) The stem group age of Jateorhiza was also constrained to be at least 47.8 Ma based on the fossil Jateorhiza gilliami Chandler [START_REF] Chandler | The Lower Tertiary Floras of southern England IV: A summary and survey of findings in the light of recent botanical observations[END_REF]. Following the suggestion of [START_REF] Parham | Best practices for justifying fossil calibrations[END_REF], we used the uniform prior distributions for these three fossil calibration points. We set a 101 Ma maximum age for the root, which is the estimated crown age of the Chasmantheroideae [START_REF] Wang | Menispermaceae and the diversification of tropical rainforests near the Cretaceous-Paleogene boundary[END_REF], with a normal distribution and a standard deviation of three. A recent study reported a relatively younger age (80.14 Ma) for the crown age of the subfamily [START_REF] Wefferling | Dated phylogeography of western NorthAmerican subalpine marshmarigolds (Caltha spp., Ranunculaceae): Miocene-Pliocene divergence of hexaploids, multiple origins of allododecaploids during the Pleistocene, and repeated recolonization of Last Glacial Maximum glaciated regions[END_REF]. To explore the impact of the root age on the estimates, we also constrained the root to 80.14 Ma and reran the dating analysis using the cpDNA data.

Both dating analyses were carried out under the GTR + I + Γ model for each marker partition separately and with a Yule tree prior. The MCMC chains were run for 50 million generations, sampling every 5000 generations. Appropriate burn-in and the effective sample size value of each parameter (>200) were assessed in Tracer v. 1.6 (Rambaut & al., 2014). The maximum clade credibility (MCC) tree with median branch lengths and 95% highest posterior density (HPD) intervals on nodes was built in TreeAnnotator v. 1.8.4 (Drummond & al., 2012).

Biogeographical analysis

To infer range evolution of Tinospora, we focused on Tinospora and its closely related genera belonging to the broad Chasmanthera . The MCC tree obtained from the cpDNA data with the root set as 101 Ma was modified in Mesquite v.2.75 [START_REF] Maddison | Mesquite, version 2.75[END_REF] and was accordingly used for biogeographical analysis in the R package BioGeoBEARS [START_REF] Matzke | Probabilistic historical biogeography: New models for founder event speciation, imperfect detection, and fossils allow improved accuracy and model-testing[END_REF]. Following [START_REF] Ree | Conceptual and statistical problems with the DEC + J model of founder-event speciation and its comparison with DEC via model selection[END_REF], we used the dispersalextinctioncladogenesis (DEC) model [START_REF] Ree | Maximum likelihood inference of geographic range evolution by dispersal, local extinction, and cladogenesis[END_REF]. Four major areas of distribution were categorized: Asia (A); Australasia (B); Africa (C); and South America (D). We allowed the inferred ancestors to occupy up to two areas (sampled tips occupy only one area). Dispersal probabilities between pairs of areas were specified based on [START_REF] Buerki | An evaluation of new parsimonybased versus parametric inferencemethods in biogeography: A case study using the globally distributed plant family Sapindaceae[END_REF].

RESULTS

Phylogenetic analyses

The cpDNA dataset comprised 7235 aligned nucleotides. ML and BI analyses yielded identical trees (Fig. 1). Calycocarpum Nutt. ex Torr. & A.Gray is the earliest-diverging lineage in Burasaieae, followed by the monophyletic group consisting of Aspidocarya Hook.f. & Thomson, Disciphania Eichler, and Parabaena , and then Tinomiscium Miers ex Hook.f. & Thomson. Tinospora tinosporoides and Borismene Barneby formed a clade with strong support (BS = 95%, PP = 1.0). Dioscoreophyllum Engl. is sister to the broad Chasmanthera (BS = 100%, PP = 1.0). Jateorhiza is sister to all other broad Chasmanthera (BS = 100%, PP = 1.0). Tinospora (excluding T. oblongifolia and T. tinosporoides) is sister to the clade I (BS = 90%, PP = 1.0). Tinospora oblongifolia (Engl.) Troupin is recovered as sister of Hyalosepalum (BS = 72%, PP = 0.96). Rhigiocarya groups with Sarcolophium (BS = 97%, PP = 1.0). Relationships among the seven clades, Borismene-Tinospora tinosporoides, Paratinospora, Penianthus longifolius, Penianthus patulinervis-Sphenocentrum jollyanum, Burasaia Thouars-Orthogynium Baillon, Fibraurea Lour., and Dioscoreophyllumbroad Chasmanthera are not resolved.

The aligned matrix of ITS sequences was 657 characters in length. ML and BI analyses produced identical trees (Fig. 2). Comparing to the cpDNA topology, the ITS topology has lower resolution, but a distant relationship between Tinospora oblongifolia and other Tinospora is also recovered. Tinospora is sister to clade II (BS = 92%, PP = 1.0). Rhigiocarya and Kolobopetalum have a sister relationship with strong support (BS = 90%, PP = 0.99).

The combined cpDNA and ITS dataset comprised 7892 aligned nucleotides. ML and BI analyses resulted in highly congruent trees (suppl. Fig. S1). The topology from the combined cpDNA and ITS dataset (suppl. Fig. S1) is congruent with the cpDNA tree (Fig. 1), but the relationships within Burasaieae are usually resolved with greater support in the combined cpDNA and ITS tree except for the placement of Rhigiocarya (the BS and PP values were 35% and 0.02 lower, respectively).

Character evolution

A summary of character states that are relevant for the taxonomy of the group is shown in Fig. 3. Exploration of character changes and ancestral-state inference were undertaken using the tree from the combined cpDNA and ITS analysis. The ML tree was chosen based on its congruence with that obtained from BI (suppl. Fig. S1). Likelihood inferences of character states indicate that all five selected morphological characters (number of sepal whorls, number of stamens, stamen connation, dehiscence types of anthers, pollen aperture types) are homoplastic in Burasaieae. For example, three sepals seem to have evolved independently at least three times (Fig. 4A). Trace of the three states for stamen connation (no, yes, and outer whorl free and inner whorl connate) appears to be extremely complex (Fig. 4B).

Divergence time estimation and biogeographical reconstruction

The different ages of the root have no effect on divergence time estimates. A comparison of estimated times from the different ages of the root is shown in Fig. 5. The age estimation using the cpDNA data with the root set as 101 Ma is here reported and was used for biogeographical analysis. Ancestral area reconstruction for Tinospora is indicated in Fig. 6. The Australian T. tinosporoides split from its South American sister Borismene at about 29.05 Ma (95% HPD: 12.43-47.44; node 1 in Fig. 5). Tinospora originated from Africa at 28.57 Ma (95% HPD: 22.67-34.76; node 2), and its most recent common ancestor had already spread to Asia by 21.54 Ma (95% HPD: 15.63-27.62; node 3). A vicariance event separated African T. tenera Miers and its Asian sister at about 17.53 Ma (95% HPD: 12.02-23.39; node 4). One dispersal from Asia to Australia occurred at about 11.16 Ma (95% HPD: 7.15-15.92; node 5). The African T. bakis (A.Rich.) Miers diverged from the ancestor of the Australian T. esiangkara and T. smilacina Benth. at about 9.71 Ma (95% HPD: 5.84-14.61; node 6).

DISCUSSION

The trees resulting from the combined cpDNA and ITS dataset are congruent with those from the cpDNA dataset (suppl. Fig. S1 vs. Fig. 1), whereas the placements of Tinospora and Rhigiocarya are incongruent in the cpDNA and ITS trees (suppl. Fig. S2). The BS value the placement of Tinospora was increased from 90% in the cpDNA data to 100% in the combined cpDNA and ITS data. A similar case for Tinospora was also reported by [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF]. Based on statistical tests, [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF] suggested that the conflict for the placement of Tinospora between the cpDNA and ITS data is artificial and resulted from high substitution rates and homoplasy in the ITS sequences. However, other causes, such as intergeneric hybridization or incomplete lineage sorting, cannot be ruled out. More nuclear DNA regions are needed to test these hypotheses in the future. For Rhigiocarya, the BS and PP values become lower in the combined cpDNA and ITS tree (Fig. 2 vs. suppl. Fig. S1). In the cpDNA tree, Rhigiocarya is sister to Sarcolophium, with which it shares longitudinally dehiscent anthers and pollen with tricolporate aperture. Conversely, the ITS data suggest Rhigiocarya as sister to Kolobopetalum, supported by outer free and inner connate stamens. Thus, Rhigiocarya might be a hybrid genus.

The intergeneric relationships within Burasaieae revealed by the cpDNA and combined cpDNA and ITS datasets are consistent with previous molecular studies [START_REF] Wang | Menispermaceae and the diversification of tropical rainforests near the Cretaceous-Paleogene boundary[END_REF][START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF][START_REF] Wefferling | Phylogeny and fruit evolution in Menispermaceae[END_REF][START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF]. As shown by [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF], the topology was consistent with the separation of Paratinospora, Hyalosepalum, and T. uviforme from Tinospora. Moreover, we found that the Tinospora species sampled in this study are distributed in three different clades (Fig. 1, suppl. Fig. S1): Tinospora tinosporoides and Borismene formed a clade; T. oblongifolia clustered with Hyalosepalum; and the remaining Tinospora species formed a clade.

The resurrection of Fawcettia and the Australasian-South American disjunction

Because a third, outermost whorl of sepals is sometimes found in three Tinospora species, T. cordifolia (Willd.) Miers ex Hook.f. & Thomson (sampled in this study), T. merrilliana Diels, and T. trilobata Diels, Forman (1981) reduced the monotypic Fawcettia to Tinospora, where it was placed as T. tinosporoides. However, our phylogenetic analyses strongly suggest that T. tinosporoides has only a distant relationshipwith Tinospora s.str. (including T. cordifolia, the type of the genus; Fig. 1). Besides the difference in number of sepal whorls, T. tinosporoides markedly differs from other Tinospora species in having triangular to ovate leaves without glandular patches below the basal nerve-axils (vs. cordate or linear leaves with glandular patches), pseudopaniculate inflorescences (vs. racemose), sessile male flowers (vs. pedicellate), densely spinulose endocarp (vs. verrucose or tuberculate), continuous, not ruminate endosperm (vs. ruminate; continuous endosperm also occurs in the New Guinean T. arfakiana Becc., a species not sampled here) [START_REF] Forman | A revision of Tinospora (Menispermaceae) in Asia to Australia and the Pacific[END_REF][START_REF] Forman | Flora of Australia[END_REF], and pollen with tricolpate aperture (vs. tricolporate) (Figs. 3,4). The sister relationship between Tinospora tinosporoides and Borismene is strongly supported (BS = 95%, PP = 1.0). They share a three-whorled calyx (Fig. 3). Nevertheless, T. tinosporoides is distinguished from Borismene in many other morphological characters, such as inflexed petals (vs. not inflexed), carpophore present (vs. absent), papery endocarps with a smooth surface (vs. bony, densely spinulose), not ruminate endosperm (vs. ruminate), and tricolpate aperture (vs. tricolporate) [START_REF] Barneby | New and notable Menispermaceae tribe Tinosporeae[END_REF][START_REF] Forman | A revision of Tinospora (Menispermaceae) in Asia to Australia and the Pacific[END_REF][START_REF] Thanikaimoni | Menispermacées: Palynologie et systématique[END_REF]. Thus, our analysis suggests that Fawcettia is a distinct genus in Burasaieae and is here resurrected.

The sister relationship between Fawcettia and Borismene recovered in this study results in an Australasian-South American genus pair. This amphi-Pacific tropical disjunction in the Southern Hemisphere is fascinating as it results in an explanatory challenge, and may shed light on the assembly of tropical floras in the past [START_REF] Thorne | Major disjunctions in the geographic ranges of seed plants[END_REF][START_REF] Givnish | Tropical intercontinental disjunctions: Gondwana breakup, immigration from the boreotropics, and transoceanic dispersal[END_REF]. Both Fawcettia and Borismene are restricted to tropical forests [START_REF] Barneby | New and notable Menispermaceae tribe Tinosporeae[END_REF][START_REF] Forman | Flora of Australia[END_REF]. Our divergence time estimates indicate that the split of Fawcettia and Borismene occurred in the middle Oligocene, ca. 29 Ma; node 1 in Fig. 5). This estimated age is partly consistent with the opening of the Drake Passage between South America and Antarctica (ca. 30-35 Ma;[START_REF] Sáez | Late Neogene lacustrine record and palaeogeography in the Quillagua-Llamara Basin, central Andean fore-arc (northern Chile)[END_REF][START_REF] Mcloughlin | The breakup history of Gondwana and its impact on pre-Cenozoic floristic provincialism[END_REF]. It has been suggested that Antarctica may have served as a migration route for some megathermal lineages between Australasia and South America in the early Tertiary [START_REF] Morley | Interplate dispersal paths for megathermal angiosperms[END_REF][START_REF] Vicente | In and out of the Neotropics: Historical biogeography of Eneopterinae crickets[END_REF]. However, during the Oligocene, global climate was cooling [START_REF] Zachos | Trends, rhythms, and aberrations in global climate 65 Ma to present[END_REF], which could have prevented migration of Menispermaceae through Antarctica. Molecular phylogenetic studies have indicated thatmigrations of megathermal lineages, such as Sapotaceae [START_REF] Bartish | Vicariance or long-distance dispersal: Historical biogeography of the pantropical subfamily Chrysophylloideae (Sapotaceae)[END_REF], Loranthaceae [START_REF] Liu | Historical biogeography of Loranthaceae (Santalales): Diversification agrees with emergence of tropical forests and radiation of songbirds[END_REF], and Anomospermeae (Menispermaceae; [START_REF] Lian | A multi-locus phylogeny for the Neotropical Anomospermeae (Menispermaceae): Implications for taxonomy and biogeography[END_REF], through Antarctica mainly occurred in the Paleocene-early Eocene, a global warming period [START_REF] Zachos | Trends, rhythms, and aberrations in global climate 65 Ma to present[END_REF]. Thus, we hypothesize that trans-Pacific long-distance dispersal may explain the modern disjunct distribution of Fawcettia and Borismene. Fruits of both genera are red and fleshy drupes and may be favored by birds [START_REF] Barneby | New and notable Menispermaceae tribe Tinosporeae[END_REF]. Trans-Pacific long-distance dispersal has been reported in some lineages, such as Coriaria L. (Coriariaceae; [START_REF] Yokoyama | Molecular phylogeny of Coriaria, with special emphasis on the disjunct distribution[END_REF], Nothofagus Blume (Nothofagaceae; [START_REF] Knapp | Relaxed molecular clock provides evidence for long-distance dispersal of Nothofagus (southern beech)[END_REF], and Heterozostera nigricaulis J.Kuo (Zosteraceae; [START_REF] Smith | Rare long-distance dispersal of a marine angiosperm across the Pacific Ocean[END_REF].

The circumscription of Hyalosepalum

Owing to the fact that the generic name Desmonema (Menispermaceae; [START_REF] Miers | On the Menispermaceae[END_REF] was first used in Euphorbiaceae by [START_REF] Rafinesque | Herbarium Rafinesquianum: Herbals, or botanical collections of C.S. Rafinesque, I[END_REF]), Troupin (1949) provided a newgeneric name, Hyalosepalum, to accommodate eight African species. However, later, [START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF] deemed that whether stamens were free or not was not enough to separate Hyalosepalum and Tinospora, and therefore reduced the former to the latter. Earlier phylogenetic analyses have indicated that T. caffra (= H. caffrum, the type of Hyalosepalum) has a distant relationship with Tinospora [START_REF] Ahmad | Phylogenetic patterns and genetic diversity of Indian Tinospora species based on chloroplast sequence data and cytochrome P450 polymorphisms[END_REF]Hoot & al., 2009;[START_REF] Jacques | Integrating fossils in a molecular-based phylogeny and testing them as calibration points for divergence time estimates in Menispermaceae[END_REF][START_REF] Wefferling | Phylogeny and fruit evolution in Menispermaceae[END_REF][START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF][START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF]. Thus, [START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF] resurrected Hyalosepalum; however, the circumscription of the resurrected genus has been challenging since only two out of the eight species of Hyalosepalum were sampled. Hyalosepalum uviforme (Baill.) Troupin (= Tinospora uviforme) has been confirmed to be a member of Chasmanthera, i.e., C. uviforme Baill. [START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF]this study). Here, we included T. oblongifolia (= H. oblongifolium (Engl.) Troupin) and T. tenera Miers (= H. tenerum (Miers) Troupin) in molecular phylogenetic analyses for the first time. Tinospora oblongifolia groups with H. caffrum with moderate support (BS = 72%, PP = 0.96; Fig. 1). The two species share three stamens with connate filaments, whereas Tinospora species have six free stamens [START_REF] Diels | Menispermaceae[END_REF][START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF]. Conversely, H. tenerum was embedded within Tinospora with strong support (Fig. 1). In agreement with this relationship is the shared presence of six stamens [START_REF] Diels | Menispermaceae[END_REF][START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF]. The filaments of H. tenerum are only connate at the base, whereas the filaments of other Hyalosepalum are completely connate.

Evolution of morphological characters

Sepal and stamen characters, including number of sepal whorls, number of stamens, stamen connation, dehiscence type of anthers, and pollen aperture type, have traditionally been used for generic recognition within Burasaieae, especially for the delimitation of Tinospora. These characters are traced on the phylogeny of the combined cpDNA and ITS data (Fig. 3). Likelihood inference indicates evolutionary patterns of morphological characters (Fig. 4). The clade containing Dioscoreophyllum and the broad Chasmanthera has two whorls of sepals, a feature that is also found in Paratinospora, Disciphania and Parabaena. Most genera of Burasaieae have six stamens. In the broad Chasmanthera, the state "three stamens" originated once, and reversals to "six stamens" occurred in Leptoterantha and Odontocarya truncata Standl., respectively. Free stamens are the plesiomorphic state of Burasaieae, and connate stamens have evolved four times independently. In Rhigiocarya, Kolobopetalum and Odontocarya truncata, the outer three stamens are free and the inner three are connate. Our phylogenetic hypothesis suggests that the dehiscence of anthers and pollen aperture types are extremely complex in the broad Chasmanthera.

The number of stamens, the presence/absence of stamen connation, dehiscence type of anthers, and pollen aperture types are homoplastic in the broad Chasmanthera, but each genus of the broad Chasmanthera has at least one unique combination of these four morphological characters except for Tinospora and Sarcolophium (Fig. 3). However, Sarcolophium is distinguished from Tinospora by having an endocarp with erect processes (vs. verruca or tubercule; [START_REF] Kessler | Menispermaceae[END_REF]. Our phylogenetic analyses indicate that there are 11 genera in the broad Chasmanthera (Fig. 1, suppl. Fig. S1). If the broad generic concept of Baillon ( 1872) is accepted, it will be extremely disruptive for nomenclature because numerous new nomenclatural combinations will be required, especially since both Tinospora and Odontocarya contain more than 30 species [START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF]. Considering phylogenetic and morphological evidence and the nomenclatural option, here we conclude that segregation of these 11 genera is strongly preferable, in agreement with earlier authors (e.g., [START_REF] Diels | Menispermaceae[END_REF][START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF][START_REF] Kessler | Menispermaceae[END_REF][START_REF] Ortiz | Phylogeny and a revised tribal classification of Menispermaceae based on molecular and morphological data[END_REF][START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF].

Biogeography of Tinospora

Our time estimates for Burasaieae are consistent with the results of previous studies [START_REF] Wang | Menispermaceae and the diversification of tropical rainforests near the Cretaceous-Paleogene boundary[END_REF][START_REF] Wang | New insights into the phylogeny of Burasaieae (Menispermaceae) with the recognition of a new genus and emphasis on the southern Taiwanese and mainland Chinese disjunction[END_REF]. This was anticipant because the same calibration fossils were used. Our results indicate that Tinospora originated from Africa in the Oligocene (28.57 Ma,; node 2 in Fig. 6), and its most recent common ancestor likely had spread into Asia in the Early Miocene (21.54 Ma,; node 3). There are four main competing hypotheses to explain floristic exchanges between the African and Asian tropics: Gondwanan vicariance or Indian raft, boreotropical migration, Miocene geodispersal, and long-distance dispersal (reviewed in [START_REF] Thomas | The historical origins of palaeotropical intercontinental disjunctions in the pantropical flowering plant family Annonaceae[END_REF]. Our estimated age for the dispersal from Africa to Asia is obviously younger than the timing of collision of India and continental Asia, ca. 50 Ma [START_REF] Van Hinsbergen | Greater India Basin hypothesis and a two-stage Cenozoic collision between India and Asia[END_REF] or 35 Ma [START_REF] Ali | Gondwana to Asia: Plate tectonics, paleogeography and the biological connectivity of the Indian sub-continent from the Middle Jurassic through latest Eocene (166-35 Ma)[END_REF], and also younger than the timing of disruption of the boreotropical forests, ca. 34 Ma [START_REF] Collinson | Vegetational and floristic changes around the Eocene/Oligocene boundary in western and central Europe[END_REF][START_REF] Wolfe | Climatic, floristic, and vegetational changes near the Eocene/Oligocene boundary in North America[END_REF][START_REF] Morley | Cretaceous and Tertiary climate change and the past distribution of megathermal rainforests[END_REF]. Thus, the Indian raft and boreotropical migration hypotheses were rejected, while the Miocene geodispersal hypothesis is temporally congruent with our inferences. During the Early to Middle Miocene, global climate was warming [START_REF] Zachos | Trends, rhythms, and aberrations in global climate 65 Ma to present[END_REF]. Moreover, due to the collision of the Afro-Arabian plate with the Iranian and Anatolian plates, land connections between Africa and Southwest Asia developed during this period [START_REF] Rögl | Palaeogeographic considerations for Mediterranean and Paratethys seaways (Oligocene to Miocene)[END_REF][START_REF] Popov | Lithological-Paleogeographic maps of Paratethys: 10 maps Late Eocene to Pliocene[END_REF]. Evidence from plant and vertebrate macrofossils and palynological data indicate that there were tropical forests in Arabia from the Early to iddle iocene andaville, rschner, 1998; [START_REF] Otero | Palaeoichthyofaunas from the lower Oligocene and Miocene of the Arabian plate: Palaeoecological and palaeobiogeographical implications[END_REF][START_REF] Ataabadi | The Miocene of western Asia: Fossil mammals at the cross-roads of faunal provinces and climate regimes[END_REF][START_REF] Pound | Global vegetation dynamics and latitudinal temperature gradients during the Mid to Late Miocene (15.97-5.33 Ma)[END_REF]. Tinospora species primarily inhabit tropical rainforests, monsoon forests, and open woodlands [START_REF] Troupin | Monographie des Menispermaceae Africaines[END_REF][START_REF] Forman | Menispermaceae[END_REF]. Thus, the Arabian corridor during the Early to Middle Miocene may have facilitated the range expansion of Tinospora. Other tropical lineages, such as Annonaceae [START_REF] Zhou | Out of-Africa' dispersal of tropical floras during the Miocene climatic optimum: Evidence from Uvaria (Annonaceae)[END_REF][START_REF] Thomas | The historical origins of palaeotropical intercontinental disjunctions in the pantropical flowering plant family Annonaceae[END_REF], Isodon (Schrad. ex Benth.) Spach (Lamiaceae; [START_REF] Yu | Phylogeny and historical biogeography of Isodon (Lamiaceae): Rapid radiation in south-west China and Miocene overland dispersal into Africa[END_REF], and Searsia F.A.Barkley (Anacardiaceae; [START_REF] Yang | Phylogenetic analyses of Searsia (Anacardiaceae) from eastern Asia and its biogeographic disjunction with its African relatives[END_REF], have also migrated through Arabia during this period.

Our analyses indicate that Tinospora was widely distributed in Africa and Asia until a vicariance event occurred in the Middle Miocene (17.53 Ma, Ma; node 4), which resulted in the divergence of the African T. tenera and its Asian allies. After the late Middle Miocene, climate deterioration occurred [START_REF] Zachos | Trends, rhythms, and aberrations in global climate 65 Ma to present[END_REF], which may have hampered the exchange of megathermal plants between Africa and Asia through Arabia. The African T. bakis originated from Asia in the Late Miocene (9.71 Ma, 5.84-14.61 Ma) with the extinction of its Asian ancestor (node 6 in Fig. 6). This age is younger than the disruption of the aforementioned Arabian Miocene corridor. Thus, transoceanic long-distance dispersal may be the most likely explanation for this colonization event. The fruits of Tinospora are brightly colored and fleshy drupes [START_REF] Forman | Menispermaceae[END_REF], which may be favored by birds. Transoceanic longdistance dispersal has been hypothesized to explain the Asian-African disjunction for other plant groups, such as Adansonia L. (Malvaceae; Baum&al., 1998), Osbeckia L. (Melastomataceae; [START_REF] Renner | Melastomataceae come full circle: Biogeographic reconstruction and molecular clock dating[END_REF], and Loranthaceae [START_REF] Liu | Historical biogeography of Loranthaceae (Santalales): Diversification agrees with emergence of tropical forests and radiation of songbirds[END_REF].

The clade including the Australian T. smilacina and T. esiangkara originated from Asia in early Late Miocene (11.16 Ma,node 5), obviously later than the convergence between Asia and New Guinea/Australian plates, ca. 25 Ma [START_REF] Morley | Palynological evidence for Tertiary plant dispersals in the Southeast Asian region in relation to plate tectonics and climate[END_REF][START_REF] Hall | Cenozoic geological and plate tectonic evolution of Southeast Asia and the Southwest Pacific: Computer-based reconstructions, model and animations[END_REF]Hall, , 2009)). This convergence led to the formation of the Wallacean transition zone between the Sunda and the Sahul shelves, and some taxa have dispersed from one side to the other [START_REF] Richardson | The Malesian floristic interchange: Plant migration patterns across Wallace's ine[END_REF]. The winteraceous fossil leaves collected from the Middle Miocene in Central Yunnan support the exchange of megathermal plants between Gondwana and Asia after the convergence (Liang & al.,201 ). The "west-to-east" migration pattern in Tinospora is also found in some other plant taxa, such as Pseuduvaria Miq. (Annonaceae; [START_REF] Su | Evolutionary divergence times in the Annonaceae: Evidence of a late Miocene origin of Pseuduvaria in Sundaland with subsequent diversification in New Guinea[END_REF], Alocasia (Schott) G.Don (Araceae; Nauheimer & al., 2012), and Begonia L. (Begoniaceae;Thomas & al., 2012).

Fig. 1 .

 1 Fig. 1. ML tree inferred from the cpDNA dataset. Numbers above branches are bootstrap values BS ≥ 50%) and posterior probabilities PP ≥ 0.60). An asteris *) indicates BS = 00 or PP = .0.

Fig. 2 .

 2 Fig. 2. ML tree inferred from the ITS dataset. Numbers above branches are bootstrap values BS ≥ 50%) and posterior probabilities PP ≥ 0.60). An asterisk (*) indicates BS = 100% or PP = 1.0.

Fig. 3 .

 3 Fig. 3. Simplified ML tree generated from the combined cpDNA and ITS data showing the distribution of selected character states. Branches in bold indicate BS ≥ 70% and PP ≥ 0.95. Characters are: number of sepal whorls (2, 3, 4), number of stamens (O, 3; •, 6; ¤, ≥9), stamen connation (Å~, no; √, yes; concentric circles, outer free and inner connate), dehiscence type of anthers (+, longitudinal; -, transversal; \, lateral), and pollen aperture type (∵, triporate; ▽, tricolpate; ▼, tricolporate; *, 6-cryptoporate; ?, missing data).

Fig. 4 .

 4 Fig. 4. Likelihood inference of character evolution in Burasaieae using Mesquite v.2.75 (Maddison & Maddison, 2011). A, Number of stamens; B, Stamen connation.

Fig. 5 .

 5 Fig. 5. Chronogram of Burasaieae based on the cpDNA data using BEAST with a 101 Ma maximum age for the root. Gray bars represent 95% highest posterior density intervals. The C1-3 indicate the locations of fossil calibration points. Nodes of interest were marked as 1-6. A comparison of estimated times for the nodes of interest is inserted in the upper left. Pal = Paleocene; Oli = Oligocene; Pl = Pliocene.

Fig. 6 .

 6 Fig. 6. Ancestral range reconstructions of the re-delimited Tinospora using BioGeoBEARS under the DEC model. Pie charts show the relative probabilities of alternative ancestral distributions, and numbers near the nodes indicate the highest ML probabilities. The node numbers correspond to those in Fig. 5. Pal = Paleocene; Pl = Pliocene.

ACKNOWLEDGMENTS

We sincerely thank Dr. Peter F. Stevens for kindly reading an earlier draft. The curators of MO and P are acknowledged for sharing plant materials. This research was partially funded by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB31030000) and the National Natural Science Foundation of China (Grant nos. 31470315, 31770231 and 31770233).

AUTHOR CONTRIBUTIONS

WW designed the research. LL performed the lab work and analyzed data. RDCO, FJ, ZDC and WW collected the materials. LL and WWinterpreted the results and wrote the paper. All authors contributed with further discussion and writing of the manuscript. -RDCO, https://orcid.org/0000-0002-4673-4347; FJ, https://orcid.org/0000-0002-7729-1067; WW, https://orcid.org/0000-0001-6901-6375