2.2 Equations of motion

The equations of motion in Eulerian form are
∂(ρu) ------= − ∇ ⋅ (ρuu ) − ∇P − ρ∇ Φ − ∇ ⋅ τvisc, (5 ) ∂t
or in Lagrangian form
Du P 1 ----= − --∇ lnP − ∇ Φ − --∇ ⋅ τvisc, (6 ) Dt ρ ρ
where P is the gas pressure, Φ is the gravitational potential, and τvisc is the viscous stress tensor.

Near the solar surface, the acceleration of gravity − ∇Φ is a nearly constant (per unit mass) downwards vertical force, which needs to be balanced by an equally large pressure gradient − Pρ∇ ln P acting in the opposite direction. Therefore, if fluid motions are sufficiently slow and large scale (horizontally), so the other terms in the equations of motion may be neglected, then what remains is a condition of hydrostatic equilibrium,

P ∂ ln P ∂Φ − -- ------= --- ≡ gz. (7 ) ρ ∂z ∂z
For constant P ∕ρ (essentially constant temperature), one finds that the logarithmic pressure ln P depends linearly on height, and that the pressure thus decreases exponentially with height,
P = P e−z∕HP, (8 ) 0
where the pressure scale height is
-P-- HP = ρg . (9 ) z
These equations, which describe an approximate hydrostatic vertical balance of forces, have some very important consequences in cases where the scale height HP is nearly independent of horizontal position (x,y), while P0 is allowed to vary slowly with horizontal position,
P0 = P0(x,y). (10 )
It then follows that the entire pressure field may vary ‘in unison’ vertically, with a common (logarithmic) variation horizontally, leading to horizontal components of the pressure gradient
∇ ⊥ lnP ≈ ∇ ⊥ ln P0(x, y), (11 )
which are essentially independent of height z. This corresponds to a smoothly undulating surface or atmosphere, where the vertical stratification is similar everywhere, except for a vertical displacement that depends only on (x,y). This is of course somewhat of an idealization, but it is nevertheless an important limiting scenario, where one can have a slowly varying horizontal velocity field that is nearly independent of height.

Returning to the anelastic approximation, ∇ ⋅ (ρu) ≈ 0, it may be combined with the equations of motion to obtain

∇2P = ∇ ⋅ [− ρu ⋅ ∇u − ρ∇ Φ − ∇ ⋅ (τvisc)], (12 )
showing that, in this approximation, the pressure is determined (instantaneously) by the per-unit-volume force field – the role of the pressure is essentially that of a constraint, enforcing the condition ∇ ⋅ (ρu) = 0.
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