Horizontal motions of air parcels are calculated using classical physics:

- (1)

- (2)

where **x** denotes horizontal position, **v** is horizontal velocity, *t* is time, *f* is the Coriolis parameter, **k** is the unit vector in the vertical, **A**_{p} is the acceleration resulting from pressure, and **A**_{m} is the acceleration resulting from turbulent mixing of momentum.

Equations (1)–(2) are ordinary differential equations that are easy to approximate using standard numerical methods once **A**_{p} and **A**_{m} are determined. To calculate **A**_{p} for every parcel in a computationally efficient manner, we assume that each parcel has the following characteristics: (1) a uniform potential temperature; (2) hydrostatic pressure; and (3) a time-invariant pressure thickness distribution *δ p*. We construct *δ p* using a bell shape *b* defined with a polynomial:

- (3)

where *δ p*_{max} is the maximum pressure thickness, *r*_{x} and *r*_{y} are the parcel radii in the and directions, the prime (′) notation denotes a coordinate system centred on the parcel, and for *d <* 1 and *b*(*d*) = 0 for *d* ≥ 1 (Figure 1(a)). Now consider the Montgomery potential (or dry static energy) *M* = *C*_{p}*T*+ *gz* where *C*_{p} is the specific heat of dry air, *T* is temperature, *g* is gravity, and *z* is height. In a hydrostatic atmosphere, the horizontal acceleration due to pressure at a particular location equals the horizontal gradient of the Montgomery potential on a constant potential temperature surface (e.g. Holton, 2004, pp 55, 109). We define the pressure acceleration of a parcel as the mass-weighted average of the gradient in the Montgomery potential across the parcel:

- (4)

where the second equality provides the form of the equation that we discretize, and the parcel weight *W* is calculated by integrating the pressure thickness over its horizontal projection *H*:

- (5)

The integrals in (4)–(5) are approximated using a Riemann sum, and the Montgomery potential is calculated following the method of Haertel *et al.* (2001). At the surface, *M* is set to *gz* (i.e. *T* is defined to be zero), and the change in the Montgomery potential (*δ M*) rising vertically across a parcel interface is given by:

- (6)

where the pressure at the interface equals the sum of pressure thickness functions above it, and temperature is calculated from potential temperature using Poisson's equation. Note that there are no approximations resulting from this vertical discretization, which is a consequence of the fact that the Montgomery potential is independent of height within an isentropic layer.

We include a vertical viscosity (0.01 m^{2} s^{−1}) to parametrize the turbulent mixing of momentum, and its implementation is identical to that in our Lagrangian ocean model (H04). Note that, for simplicity, we introduce the equations for the LAM in Cartesian coordinates, but they are adapted to spherical coordinates following the method described by H04 for applications on the aquaplanet (see below). Finally, we mention that the number of computations required to solve (1)–(6) scales linearly with the number of parcels. This is also the case for our Lagrangian ocean model, which has been found to be computationally competitive with Eulerian ocean models when run at a similar resolution (H04).