DQM/Dt is a material derivative, which means the rate of change is taken along the motion of the infinitesimal volume element. Local heating of an infinitesimal volume element can be due to several sources. We list a few of them here.

Heating by conduction At the molecular level the heat exchange from one infinitesimal volume element to its neighbors with differing temperature is given by the divergence of the heat flux V ■ h(r, t). This gives the cooling rate per unit volume of the moving element. To obtain the cooling rate per unit mass, one must divide by density p(r, t).

Heating by phase change As a moist parcel moves it might experience a temperature change and this could lead to condensation (or evaporation) onto droplets. The release of enthalpy is given by L dMvap = MailL dws, where ws(T) is the saturation mixing ratio of the volume element containing Mair of air. Hence, the heating rate per unit mass is simply -L dws (note that dws is negative for condensation).

Heating by radiation In this case we have a certain radiation flux density of energy F(r, t). The heating rate per unit volume is -V ■ F(r, t). And the heating rate per unit mass is —(1/p)V -F(r, t).

To summarize we have

DQm 1 Dws 1

Sometimes a frictional heating term is included as well. Generally in applications such as numerical weather forecasting and climate modeling, the first term above is small compared to the others and is neglected.

Notes

This chapter is really an introduction to dynamics. Most dynamics books cover these subjects and many do so in more detail, see for example, Holton (1992).

Notation and abbreviations for Chapter 9 | |

a, b |
etc. arbitrary vectors |

a, ax |
these are used for vector acceleration and its x |

component | |

ax, ay, a^z |
the Cartesian components of vector a |

A |
a vector field |

div2V |
divergence of vector field V in two dimensions |

D/Dt |
material derivative |

DQu/Dt |
the rate of heating of a moving parcel per unit mass |

(Js-1 kg-1) | |

V ■ F(r, t) |
divergence of the vector field F(r, t) |

V T = (d T /d x)i + ■ ■ |
gradient of T |

i, j, k |
the Cartesian unit vectors |

kh |
thermal conductivity (JK-1 m-1 s-1) |

L |
enthalpy of evaporation (latent heat) (J kg-1) |

n |
unit vector |

ne |
unit vector in the theta direction (polar coordinates) |

d T/d n |
directional derivative of T |

r = xi + yj + zk |
position vector |

r |
unit vector in the r direction (polar coordinates) |

S, dS surface, surface element vector v three-dimensional velocity vector field (ms-1)

V upper case usually indicates two-dimensional velocity

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