The hydrostatic multilayer solver for free-surface flows

    The theoretical basis and main algorithms for this solver are described in Popinet, 2020. Note however that this version of the solver is more recent and may not match the details of Popinet, 2020.

    The system of n layers of incompressible fluid pictured below is modelled as the set of (hydrostatic) equations: \displaystyle \begin{aligned} \partial_t h_k + \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \right)_k & = 0,\\ \partial_t \left( h \mathbf{u} \right)_k + \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \mathbf{u} \right)_k & = - gh_k \mathbf{{\nabla}} (\eta) \end{aligned} with \mathbf{u}_k the velocity vector, h_k the layer thickness, g the acceleration of gravity and \eta the free-surface height. The non-hydrostatic pressure \phi_k and vertical velocity w_k can be added using the non-hydrostatic extension. See also the general introduction.

    Definition of the n layers.

    Definition of the n layers.

    Fields and parameters

    The zb and eta fields define the bathymetry and free-surface height. The h and u fields define the layer thicknesses and velocities.

    The acceleration of gravity is G and dry controls the minimum layer thickness. The hydrostatic CFL criterion is defined by CFL_H. It is set to a very large value by default, but will be tuned either by the user or by the default solver settings (typically depending on whether time integration is explicit or implicit).

    The gradient pointer gives the function used for limiting.

    tracers is a list of tracers for each layer. By default it contains only the components of velocity (unless linearised is set to true in which case the tracers list is empty).

    #define BGHOSTS 2
    #define LAYERS 1
    #include "utils.h"
    scalar zb[], eta, h;
    vector u;
    double G = 1., dry = 1e-12, CFL_H = 1e40;
    double (* gradient) (double, double, double) = minmod2;
    scalar * tracers = NULL;
    bool linearised = false;


    We first define refinement and restriction functions for the free-surface height eta which is just the sum of all layer thicknesses and of bathymetry.

    #if TREE
    static void refine_eta (Point point, scalar eta)
      foreach_child() {
        eta[] = zb[];
          eta[] += h[];
    static void restriction_eta (Point point, scalar eta)
      eta[] = zb[];
        eta[] += h[];
    #endif // TREE

    The allocation of fields for each layer is split in two parts, because we need to make sure that the layer thicknesses and \eta are allocated first (in the case where other layer fields are added, for example for the non-hydrostatic extension).

    event defaults0 (i = 0)
      assert (nl > 0);
      h = new scalar[nl];
      h.gradient = gradient;
    #if TREE
      h.refine = h.prolongation = refine_linear;
      h.restriction = restriction_volume_average;
      eta = new scalar;
      reset ({h, zb}, 0.);

    We set the proper gradient and refinement/restriction functions.

      zb.gradient = gradient;
      eta.gradient = gradient;
    #if TREE
      zb.refine = zb.prolongation = refine_linear;
      zb.restriction = restriction_volume_average;
      eta.prolongation = refine_linear;
      eta.refine  = refine_eta;
      eta.restriction = restriction_eta;
    #endif // TREE
    #include "run.h"

    Other fields, such as \mathbf{u}_k here, are added by this event.

    event defaults (i = 0)

    The (velocity) CFL is limited by the unsplit advection scheme, so is dependent on the dimension. The (gravity wave) CFL is set to 1/2 (if not already set by the user).

      CFL = 1./(2.*dimension);
      if (CFL_H == 1e40)
        CFL_H = 0.5;  
      u = new vector[nl];
      reset ({u}, 0.);
      if (!linearised)
          tracers = list_append (tracers, u.x);

    The gradient and prolongation/restriction functions are set for all tracer fields.

      for (scalar s in tracers) {
        s.gradient = gradient;
    #if TREE
        s.refine = s.prolongation = refine_linear;
        s.restriction = restriction_volume_average;

    We setup the default display.

      display ("squares (color = 'eta > zb ? eta : nodata', spread = -1);");

    After user initialisation, we define the free-surface height \eta.

    event init (i = 0)
      foreach() {
        eta[] = zb[];
          eta[] += h[];
      boundary (all);

    Computation of face values

    At each timestep, temporary face fields are defined for the fluxes (h \mathbf{u})^{n+1/2}, face height h_f^{n+1/2} and height-weighted face accelerations (ha)^{n+1/2}.

    #define gmetric(i) (2.*fm.x[i]/(cm[i] + cm[i-1]))
    static bool hydrostatic = true;
    face vector hu, hf, ha;
    event face_fields (i++, last)
      hu = new face vector[nl];
      hf = new face vector[nl];
      ha = new face vector[nl];

    The (CFL-limited) timestep is also computed by this function. A difficulty is that the prediction step below also requires an estimated timestep (the pdt variable below). The timestep at the previous iteration is used as estimate. For the initial timestep a “sufficiently small” value is used.

      static double pdt = 1e-6;
      double dtmax = DT;
      foreach_face (reduction (min:dtmax)) {
        double ax = - gmetric(0)*G*(eta[] - eta[-1])/Delta;
        double H = 0., um = 0.;
        foreach_layer() {

    The face velocity is computed as the height-weighted average of the cell velocities.

          double hl = h[-1] > dry ? h[-1] : 0.;
          double hr = h[] > dry ? h[] : 0.;
          hu.x[] = hl > 0. || hr > 0. ? (hl*u.x[-1] + hr*u.x[])/(hl + hr) : 0.;

    The face height is computed using a variant of the BCG scheme.

          double hff, un = pdt*(hu.x[] + pdt*ax)/Delta, a = sign(un);
          int i = - (a + 1.)/2.;
          double g = h.gradient ? h.gradient (h[i-1], h[i], h[i+1])/Delta :
    	(h[i+1] - h[i-1])/(2.*Delta);
          hff = h[i] + a*(1. - a*un)*g*Delta/2.;
          hf.x[] = fm.x[]*hff;

    The maximum velocity is stored and the flux and height-weighted accelerations are computed.

          if (fabs(hu.x[]) > um)
    	um = fabs(hu.x[]);
          hu.x[] *= hf.x[];
          ha.x[] = hf.x[]*ax;
          H += hff;

    The maximum timestep is computed using the total depth H and the advection and gravity wave CFL criteria. The gravity wave speed takes dispersion into account in the non-hydrostatic case.

        if (H > dry) {
          double c = um/CFL + (hydrostatic ? sqrt(G*H) :
          double dt = min(cm[], cm[-1])*Delta/(c*fm.x[]);
          if (dt < dtmax)
    	dtmax = dt;
      boundary ((scalar *){ha, hu, hf});

    The timestep is computed, taking into account the timing of events, and also stored in pdt (see comment above).

      pdt = dt = dtnext (dtmax);

    Advection and diffusion

    The function below approximates the advection terms using estimates of the face fluxes h\mathbf{u} and face heights h_f.

    void advect (scalar * tracers, face vector hu, face vector hf, double dt)

    The fluxes are first limited according to the CFL condition to ensure strict positivity of the layer heights. This step is necessary due to the approximate estimation of the CFL condition in the timestep calculation above.

        foreach_layer() {
        double hul = hu.x[];
        if (hul*dt/(Delta*cm[-1]) > CFL*h[-1])
          hul = CFL*h[-1]*Delta*cm[-1]/dt;
        else if (- hul*dt/(Delta*cm[]) > CFL*h[])
          hul = - CFL*h[]*Delta*cm[]/dt;

    In the case where the flux is limited, and for multiple layers, an attempt is made to conserve the total barotropic flux by merging the flux difference with the flux in the layer just above. This allows to maintain an accurate evolution for the free-surface height \eta.

        if (hul != hu.x[]) {
          if (point.l < nl - 1)
    	hu.x[0,0,1] += hu.x[] - hul;
          else if (nl > 1)
    	fprintf (stderr, "warning: could not conserve barotropic flux "
    		 "at %g,%g,%d\n", x, y, point.l);
          hu.x[] = hul;
      boundary ((scalar *){hu});
      face vector flux[];
      foreach_layer() {

    We compute the flux (shu)_{i+1/2,k} for each tracer s, using a variant of the BCG scheme.

        for (scalar s in tracers) {
          foreach_face() {
    	double un = dt*hu.x[]/(hf.x[]*Delta + dry), a = sign(un);
    	int i = -(a + 1.)/2.;
    	double g = s.gradient ?
    	  s.gradient (s[i-1], s[i], s[i+1])/Delta :
    	  (s[i+1] - s[i-1])/(2.*Delta);
    	double s2 = s[i] + a*(1. - a*un)*g*Delta/2.;
    #if dimension > 1
    	if (hf.y[i] + hf.y[i,1] > dry) {
    	  double vn = (hu.y[i] + hu.y[i,1])/(hf.y[i] + hf.y[i,1]);
    	  double syy = (s.gradient ? s.gradient (s[i,-1], s[i], s[i,1]) :
    			vn < 0. ? s[i,1] - s[i] : s[i] - s[i,-1]);
    	  s2 -= dt*vn*syy/(2.*Delta);
    #endif // dimension > 1
    	flux.x[] = s2*hu.x[];
          boundary_flux ({flux});

    We compute (hs)^\star_i = (hs)^n_i + \Delta t [(shu)_{i+1/2} -(shu)_{i-1/2}]/\Delta.

          foreach() {
    	s[] *= h[];
    	  s[] += dt*(flux.x[] - flux.x[1])/(Delta*cm[]);

    We then obtain h^{n+1} and s^{n+1} using \displaystyle \begin{aligned} h_i^{n+1} & = h_i^n + \Delta t \frac{(hu)_{i+1/2} - (hu)_{i-1/2}}{\Delta},\\ s_i^{n+1} & = \frac{(hs)^\star_i}{h_i^{n+1}} \end{aligned}

        foreach() {
          double h1 = h[];
    	h1 += dt*(hu.x[] - hu.x[1])/(Delta*cm[]);
          if (h1 < - 1e-12)
    	fprintf (stderr, "warning: h1 = %g < - 1e-12 at %g,%g,%d,%g\n",
    		 h1, x, y, _layer, t);
          h[] = fmax(h1, 0.);
          if (h1 < dry) {
    	for (scalar f in tracers)
    	  f[] = 0.;
    	for (scalar f in tracers)
    	  f[] /= h1;      
      scalar * list = list_copy (tracers);
      list = list_append (list, h);
      boundary (list);
      free (list);

    This is where the ‘two-step advection’ of the implicit scheme plugs itself (nothing is done for the explicit scheme).

    event half_advection (i++, last);

    Vertical diffusion (including viscosity) is added by this code.

    #include "diffusion.h"


    Acceleration terms are added here. In the simplest case, this is only the pressure gradient due to the free-surface slope, as computed in face_fields.

    event acceleration (i++, last);
    event pressure (i++, last)

    The acceleration is applied to the face fluxes…

          hu.x[] += dt*ha.x[];
      boundary ((scalar *){hu});

    … and to the centered velocity field, using height-weighting.

        foreach_layer() {
    	u.x[] += dt*(ha.x[] + ha.x[1])/(hf.x[] + hf.x[1] + dry);
    #if dimension == 2
          // metric terms
          double dmdl = (fm.x[1,0] - fm.x[])/(cm[]*Delta);
          double dmdt = (fm.y[0,1] - fm.y[])/(cm[]*Delta);
          double ux = u.x[], uy = u.y[];
          double fG = uy*dmdl - ux*dmdt;
          u.x[] += dt*fG*uy;
          u.y[] -= dt*fG*ux;
    #endif // dimension == 2
      boundary ((scalar *){u});
      delete ((scalar *){ha});

    The resulting fluxes are used to advect both tracers and layer heights.

      advect (tracers, hu, hf, dt);

    Finally the free-surface height \eta is updated and the boundary conditions are applied.

    event update_eta (i++, last)
      delete ((scalar *){hu, hf});
      foreach() {
        double etap = zb[];
          etap += h[];
        eta[] = etap;
      boundary ({eta});

    Vertical remapping and mesh adaptation

    Vertical remapping is applied here if necessary.

    event remap (i++, last);
    #if TREE
    event adapt (i++,last);


    The fields and lists allocated in defaults() above must be freed at the end of the run.

    event cleanup (t = end, last)
      delete ({eta, h, u});
      free (tracers), tracers = NULL;

    Horizontal pressure gradient

    The macro below computes the horizontal pressure gradient \displaystyle pg_k = - \mathbf{{\nabla}} (h \phi)_k + \left[ \phi \mathbf{{\nabla}} z \right]_k on the faces of each layer. The slope of the layer interfaces \mathbf{{\nabla}} z_{k+1/2} in the second-term is bounded by max_slope (by default 30 degrees).

    double max_slope = 0.577350269189626; // = tan(30.*pi/180.)
    #define slope_limited(dz) (fabs(dz) < max_slope ? (dz) :	\
    			   ((dz) > 0. ? max_slope : - max_slope))
    #define hpg(pg,phi,i) {							\
      double dz = zb[i] - zb[i-1];						\
      foreach_layer() {							\
        pg = 0.;								\
        if (h[i] + h[i-1] > dry) {						\
          double s = Delta*slope_limited(dz/Delta);				\
          pg -= (h[i] + s)*phi[i] - (h[i-1] - s)*phi[i-1];			\
          if (point.l < nl - 1) {						\
    	double s = Delta*slope_limited((dz + h[i] - h[i-1])/Delta);     \
    	pg -= (h[i] - s)*phi[i,0,1] - (h[i-1] + s)*phi[i-1,0,1];	\
          }									\
          pg *= gmetric(i)*hf.x[i]/(Delta*(h[i] + h[i-1]));			\
    #define end_hpg(i)				\
        dz += h[i] - h[i-1];			\
      }						\

    Hydrostatic vertical velocity

    For the hydrostatic solver, the vertical velocity is not defined by default since it is usually not required. The function below can be applied to compute it using the (diagnostic) incompressibility condition \displaystyle \mathbf{{\nabla}} \cdot \left( h \mathbf{u} \right)_k + \left[ w - \mathbf{u} \cdot \mathbf{{\nabla}} (z) \right]_k = 0

    void vertical_velocity (scalar w, face vector hu, face vector hf)
      foreach() {
        double dz = zb[1] - zb[-1];
        double wm = 0.;
        foreach_layer() {
          w[] = wm + (hu.x[] + hu.x[1])/(hf.x[] + hf.x[1] + dry)*
    	(dz + h[1] - h[-1])/(2.*Delta);
          if (point.l > 0)
    	  w[] -= (hu.x[0,0,-1] + hu.x[1,0,-1])
    	  /(hf.x[0,0,-1] + hf.x[1,0,-1] + dry)*dz/(2.*Delta);
    	w[] -= (hu.x[1] - hu.x[])/Delta;
          dz += h[1] - h[-1], wm = w[];
      boundary ({w});

    “Radiation” boundary conditions

    This can be used to implement open boundary conditions at low Froude numbers. The idea is to set the velocity normal to the boundary so that the water level relaxes towards its desired value (ref).

    double _radiation (Point point, double ref, scalar s)
      double H = 0.;
        H += h[];
    #if 0  
      return H > dry ? sqrt(G/H)*(zb[] + H - ref) : 0.;
      return sqrt (G*max(H,0.)) - sqrt(G*max(ref - zb[], 0.));
    #define radiation(ref) _radiation(point, ref, _s)

    Conservation of water surface elevation

    We re-use some generic functions.

    #include "elevation.h"

    But we need to re-define the water depth refinement function.

    #if TREE
    static void refine_layered_elevation (Point point, scalar h)

    We first check whether we are dealing with “shallow cells”.

      bool shallow = zb[] > default_sea_level;
        if (zb[] > default_sea_level)
          shallow = true, break;

    If we do, refined cells are just set to the default sea level.

      if (shallow)
          h[] = max(0., default_sea_level - zb[]);

    Otherwise, we use the surface elevation of the parent cells to reconstruct the water depth of the children cells.

      else {
        double eta = zb[] + h[];   // water surface elevation  
        coord g; // gradient of eta
        if (gradient)
    	g.x = gradient (zb[-1] + h[-1], eta, zb[1] + h[1])/4.;
    	g.x = (zb[1] + h[1] - zb[-1] - h[-1])/(2.*Delta);
        // reconstruct water depth h from eta and zb
        foreach_child() {
          double etac = eta;
    	etac += g.x*child.x;
          h[] = max(0., etac - zb[]);

    We overload the conserve_elevation() function.

    void conserve_layered_elevation (void)
      h.refine  = refine_layered_elevation;
      h.prolongation = prolongation_elevation;
      h.restriction = restriction_elevation;
      boundary ({h});
    #define conserve_elevation() conserve_layered_elevation()
    #endif // TREE
    #include "gauges.h"
    #if dimension == 2

    Fluxes through sections

    These functions are typically used to compute fluxes (i.e. flow rates) through cross-sections defined by two endpoints (i.e. segments). Note that the orientation of the segment is taken into account when computing the flux i.e the positive normal direction to the segment is to the “left” when looking from the start to the end.

    This can be expressed mathematically as: \displaystyle \text{flux}[k] = \int_A^B h_k\mathbf{u}_k\cdot\mathbf{n}dl with A and B the endpoints of the segment, k the layer index, \mathbf{n} the oriented segment unit normal and dl the elementary length. The function returns the sum (over k) of all the fluxes.

    double segment_flux (coord segment[2], double * flux, scalar h, vector u)
      coord m = {segment[0].y - segment[1].y, segment[1].x - segment[0].x};
      normalize (&m);
      for (int l = 0; l < nl; l++)
        flux[l] = 0.;
      foreach_segment (segment, p) {
        double dl = 0.;
        foreach_dimension() {
          double dp = (p[1].x - p[0].x)*Delta/Delta_x*(fm.y[] + fm.y[0,1])/2.;
          dl += sq(dp);
        dl = sqrt (dl);    
        for (int i = 0; i < 2; i++) {
          coord a = p[i];
    	flux[_layer] += dl/2.*
    	interpolate_linear (point, (struct _interpolate)
    			    {h, a.x, a.y, 0.})*
    	(m.x*interpolate_linear (point, (struct _interpolate)
    				 {u.x, a.x, a.y, 0.}) +
    	 m.y*interpolate_linear (point, (struct _interpolate)
    				 {u.y, a.x, a.y, 0.}));
      // reduction
    #if _MPI
      MPI_Allreduce (MPI_IN_PLACE, flux, nl, MPI_DOUBLE, MPI_SUM, MPI_COMM_WORLD);
      double tot = 0.;
      for (int l = 0; l < nl; l++)
        tot += flux[l];
      return tot;

    A NULL-terminated array of Flux structures passed to output_fluxes() will create a file (called name) for each flux. Each time output_fluxes() is called a line will be appended to the file. The line contains the time, the total flux and the value of the flux for each h, u pair in the layer. The desc field can be filled with a longer description of the flux.

    typedef struct {
      char * name;
      coord s[2];
      char * desc;
      FILE * fp;
    } Flux;
    void output_fluxes (Flux * fluxes, scalar h, vector u)
      for (Flux * f = fluxes; f->name; f++) {
        double flux[nl];
        double tot = segment_flux (f->s, flux, h, u);
        if (pid() == 0) {
          if (!f->fp) {
    	f->fp = fopen (f->name, "w");
    	if (f->desc)
    	  fprintf (f->fp, "%s\n", f->desc);
          fprintf (f->fp, "%g %g", t, tot);
          for (int i = 0; i < nl; i++)
    	fprintf (f->fp, " %g", flux[i]);
          fputc ('\n', f->fp);
          fflush (f->fp);
    #endif // dimension == 2