Astron. Astrophys. 329, 559-570 (1998)

2. Method of numerical calculations

We consider a geometrically thin and axi-symmetric disk. This assumption allows a separate treatment of the vertical (local) and radial (global) structure of the disk. This standard procedure to calculate a time-dependent disk is described in e.g. Ichikawa & Osaki (1992).

2.1. Basic equations for the radial structure

The basic equations governing the radial (global) structure for vertically integrated quantities in a disk are the equation of continuity, the Navier-Stokes equations for the radial flow velocity and the azimuthal velocity , and the energy equation (see e.g. Bird et al. 1966). Though in thin disks the deviations from Kepler rotation are always very small their gradients in the steep transition fronts become of the same order as the gradients of the Kepler rotation themselves. They thus affect the frictional heating and angular momentum transport within the fronts. The evolution of such non-Keplerian disks is investigated here for the first time. In the vertically integrated form we use the vertically integrated quantities surface density , pressure , -component of the viscous stress tensor and the midplane values temperature T and mean molecular weight . This is a generally used approximation (see Mineshige & Osaki 1983, Cannizzo 1993).

The equation of continuity is

with r distance from the white dwarf.

The Navier-Stokes equation for the azimuthal velocity is

with , the angular momentum per unit area,

the viscous stress tensor, where f denotes the viscosity integral (see Eq. 13) and the angular velocity.

The Navier-Stokes equation for the radial flow velocity is

where is the momentum per unit area in radial direction and

the gravitational acceleration in radial direction, with G the gravitational constant, the mass of the white dwarf, and

the pressure ( gas constant, a radiation density constant). H is the isothermal scale-height

with the Keplerian angular velocity.

The vertically integrated components of the viscous stress tensor are given by (compare Ichikawa & Osaki 1994)

where is the vertically integrated bulk viscosity. Here we take and note that terms of the same form are already included in the equations. It would not be appropriate to choose as is sometimes done in physical literature in other contexts since here lateral compression involves vertical expansion and therefore (shear) viscosity.

The energy equation is

The specific heat at constant volume , the Rosseland mean opacity , and the kinematic viscosity are mid-plane values. The relation is used to determine the density at the mid-plane.

For the terms at the right hand side of the energy equation we proceed as follows. The relation between the effective temperature and the mid-plane temperature, the radiative cooling term , is obtained from integration of the vertical structure as described in Sect. 2.2. For the viscous heating we have also considered, besides the "standard" -heating term, the - and -contributions. The last two terms in the energy equation represent the radial energy flux carried by radiative and by viscous processes (compare Taam & Lin 1984, Mineshige 1987, Cannizzo 1993). For optically thin regions, the radiative transport in radial direction is neglected.

In addition to the 4 basic equations we need a description of the viscosity. Godon (1995) has shown, how the standard alpha viscosity prescription (Shakura & Sunyaev 1973, Novikov & Thorne 1973) has to be modified if the rotation law of the disk is not Keplerian. For our calculations we used the parametrisation of the viscosity given by Godon:

where is the viscosity parameter, the sound velocity, and

For this reduces to the standard Shakura-Sunyaev description. For the viscosity integral f we make use of a one-zone model for the vertical structure:

where is taken as the value of in the mid-plane. With and Eq. (11) we obtain:

We take the two-alpha description, in order to fit observed light curves (see Sect. 2.3).

2.2. The approximation of the cooling function

To solve the system of differential equations the vertically integrated quantities have to be expressed as functions of and T at each distance r. This involves a relation between and T (mid-plane), . Detailed results for vertical disk structure are given in Ludwig et al. (1994). The S-shaped thermal equilibrium curve consits of the hot branch (ionized matter), an intermediate branch (partial ionization) and a cool branch (unionized matter), optically thick or thin (compare Meyer-Hofmeister & Ritter 1992). For our calculations we use partly results of Ludwig et al. (1994) and partly, for simplicity, relations optained by Ichikawa & Osaki (1992), who consider only an optically thin cool branch, and by Cannizzo (1993).

2.2.1. The hot branch

We use the relations from Cannizzo (1993):

and

where , , and .

The relation for the effective temperature is obtained by eliminating by means of Eq. 15from Eq. 16,

where and .

2.2.2. The unstable intermediate branch

Suffix A and B denote values at the transition from the hot branch to the unstable branch and that from the unstable branch to the cool optically thick branch respectively. From our calculations of the vertical structure we obtain (see Ludwig et al. 1994)

where . For the mid-plane temperature at these points we get

Similar relations were obtained by Cannizzo (1993) and Ichikawa & Osaki (1992). The effective temperature for the unstable branch is approximated by the interpolation formula

2.2.3. The cool optically thin branch

The cool branch includes optically thick structure for higher temperatures (see Sect. 2.2.4) and optically thin structure for low temperatures. For the cool optically thin branch we use the formula given by Ichikawa & Osaki (1992). For convenience we have defined an effective temperature for the optically thin state by setting the radiative flux F equal . We get

2.2.4. The cool optically thick branch

This branch is approximated analoguously to the unstable intermediate branch. For the point T, which marks the transition from the cool optically thick to the cool optically thin state, we obtain with Eq. (25)

Dissociation of molecules and the resulting change of can lead, for some parameter combinations, to a negative slope of this branch. Such a disk structure is thermally and diffusively unstable. The effective temperature on the cool optically thick branch is given by interpolation between points B and T,

With Eqs. 17, 24, 25and 28the cooling function is completely defined.

2.3. The solution of the partial differential equations

The partial differential equations are solved by using the method of finite-differences with a time-explicit, multi-step solution procedure (see Stone & Norman 1992, Müller 1994). The accretion disk is divided into concentric rings of finite width, which are numbered from 1 to N. Number 1 is the innermost and number N the outermost ring. For the inner and outer boundary condition two "ghost-rings" with numbers 0 and N+1 are formally added. Each concentric ring has an inner and outer boundary, which are numbered as follows. The inner boundary of the i-th ring with i-1, the outer boundary with i. The variables surface density, mid-plane temperature, and azimuthal velocity are assigned at the center of rings, while the radial flow velocity is assigned on the interfaces (for illustration see Fig. 1).

Fig. 1. Illustration of the discretisation of the one-dimensional computational domain.

The radial coordinate of the center of the ring number 0 is equal to the radius of the white dwarf. At this inner boundary we assume and . At the center of the rings number 0 and 1 the azimuthal velocity is taken as Keplerian. With Keplerian velocity the equation of angular momentum conservation yields:

This equation is used for calculating the radial momentum at the interface between the rings number 0 and 1. The outer boundary is treated similarly: at the center of the rings N and N+1 the azimuthal velocity is set equal to the Keplerian flow velocity. For the radial coordinate of the center of the ring N we take 70 percent of the primary's critical Roche radius . At the interface N a constant mass transfer rate is taken. The temperature at the center of the ghost-ring N+1 is the same as that of the ring N. This means that the temperature gradient at the interface N is zero. The assumption of Kepler velocity at the inner and outer boundary and the asumption of zero temperature gradient at the outer boundary are made for definitenes. Physical situations in real disks might differ from this. One expects, however, that this differences "gets lost" a few grid points away from the boundary.

As already mentioned, we use a two-alpha description as has been shown necessary to fit observed light curves in all computations for dwarf nova outbursts. Here, the alpha-value for each ring is defined in the following way:

where for our calculations we have set and .

Since we use an explicit method for the time integration, the validity of the Courant-Friedrichs-Lewi-condition is checked at each time step in order to guarantee numerical stability.

© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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