Structure and dynamical characteristics of mid-latitude fronts

Front: A boundary whose primary structural and dynamical characteristic is a larger then background density (temperature) contrast A zero-order front: A front characterized by a discontinuity in temperature and density This type of front does not occur in the atmosphere, but does exist where two fluids of different density approach one another as illustrated below mixing associated with friction prevents atmospheric fronts from becoming zero-order ATMOSPHERIC FRONTS Gradients in temperature and density are discontinuous across fronts

Let’s for the moment consider a zero-order front We will assume that: 1) pressure must be continuous across the front 2) front is parallel to x axis 3) front is steady-state

Warm side of front Cold side of front Substitute hydrostatic equation and equate expressions: Solve for the slope of the front

For cold air to underlie warm air, slope must be positive Therefore: 1) Across front pressure gradient on the cold side must be larger that the pressure gradient on the warm side Substituting geostrophic wind relationship 2) Front must be characterized by positive geostrophic relative vorticity

Real (first order) fronts 1)Larger than background horizontal temperature (density) contrasts 2)Larger than background relative vorticity 3)Larger than background static stability

Working definition of a cold or warm front The leading edge of a transitional zone that separates advancing cold (warm) air from warm (cold) air, the length of which is significantly greater than its width. The zone is characterized by high static stability as well as larger-than- background gradients in temperature and relative vorticity.

EXAMPLES OF FRONTS

Frontogenesis Ageostrophic Circulations associated with fronts and jetstreaks

The formation of a front is called frontogenesis The decay of a front is called frontolysis These processes are described quantitatively in terms of the Three-Dimensional Frontogenesis Function Where is the magnitude of the 3-D potential temperature gradient and the total derivative implies that the change in the  gradient is calculated following air-parcel motion

The processes by which a front forms or decays can be understood more directly by expanding the frontogenetical function Algebraically, this involves expanding the total derivative expanding the term involving the magnitude of the gradient Reversing the order of differentiation, differentiating, and then using the thermodynamic equation to replace the term in the resulting equation.

( The Three-Dimensional Frontogenesis Function ) The solution becomes

( ) The terms in the yellow box all contain the derivative which is the diabatic heating rate. These terms are called the diabatic terms.

( ) The terms in this yellow box represent the contribution to frontogenesis due to horizontal deformation flow.

( ) The terms in this yellow box represent the contribution to frontogenesis due to vertical shear acting on a horizontal temperature gradient.

( ) The terms in this yellow box represent the contribution to frontogenesis due to tilting.

( ) The term in this yellow box represents the contribution to frontogenesis due to divergence.

( ) Weighting factor Adjustment for specific heat of air and air pressure Horizontal gradient in diabatic heating or cooling rate Magnitude of  gradient in one direction Magnitude of total  gradient

Gradient in diabatic heating in x direction Gradient in diabatic heating in y direction Can you think of other examples where this term might be important to frontogenesis?

( ) Weighting factor Adjustment for specific heat of air Vertical gradient in diabatic heating or cooling rate adjusted for pressure altitude Magnitude of  gradient in one direction Magnitude of total  gradient

    

( ) Stretching deformation Shearing deformation Stretching Deformation Weighting factors Deformation acting on temperature gradient Deformation acting on temperature gradient Magnitude of  gradient in one direction Magnitude of total  gradient

x y x y Time = t Time = t +  t T T-  T- 2  T T- 3  T T- 4  T T- 5  T T- 6  T T- 7  T T- 8  T T T-  T- 2  T T- 3  T T- 4  T T- 5  T T- 6  T T- 7  T T- 8  T Stretching Deformation

( ) Stretching deformation Shearing deformation Shearing Deformation Weighting factors Magnitude of  gradient in one direction Magnitude of total  gradient Deformation acting on temperature gradient Deformation acting on temperature gradient

x y T T-  T- 2  T T- 3  T T- 4  T T- 5  T T- 6  T T- 7  T T- 8  T x y T T-  T- 2  T T- 3  T T- 4  T T- 5  T T- 6  T T- 7  T T- 8  T Shearing Deformation

( ) Vertical shear acting on a horizontal temperature gradient (also called vertical deformation term) Weighting factor Magnitude of  gradient in one direction Magnitude of total  gradient Vertical shear of E-W wind Component acting on a horizontal temp gradient in x direction Vertical shear of N-S wind component acting on a horizontal temp gradient in y direction

Vertical shear acting on a horizontal temperature gradient     Before x z z x After

( ) Tilting terms Weighting factor Magnitude of  gradient in one direction Magnitude of total  gradient Tilting Of vertical  Gradient (E-W direction) Tilting Of vertical  Gradient (N-S direction)

Tilting terms    Before    After x or y z z

( ) Differential vertical motion (also called divergence term because  w/  z is related to divergence through continuity equation) Weighting factor Magnitude of  gradient in one direction Magnitude of total  gradient Compression of vertical  Gradient by differential vertical motion

Differential vertical motion    Before x or y z    After x or y z

Another view of the 2D frontogenesis function Recall the kinematic quantities: divergence (D) vorticity (  ) stretching deformation (F 1 ) shearing deformation (F1). and note that: Substituting:

This expression can be reduced to: x y x y Shearing and stretching deformation “look alike” with axes rotated

We can simplify the 2D frontogenesis equation by rotating our coordinate axes to align with the axis of dilitation of the flow (x´)

This equation illustrates that horizontal frontogenesis is only associated with divergence and deformation, but not vorticity

Yet another view of the 2D frontogenesis function Let’s replace u and v with their geostrophic components and examine geostrophic frontogenesis: Recalling the Q vector Therefore:

Magnitude of geostrophic frontogenesis is a scalar multiple of the cross isentropic component of the Q vector Convergence of Q vectors associated with rising motion Implication: Direct circulation (warm air rising and cold air sinking) associated with frontogenesis Divergence of Q vectors associated with descending motion

Is geostrophic frontogenesis, as represented by the Q vector, sufficient to describe the circulation about a front? Consider a simple north-south front undergoing frontogenesis by the geostrophic wind Assume that the confluence occurs at a constant rate k integrate to get: Using typical values of it takes 10 5 seconds or about 1 day for geostrophic confluence to increase the temperature gradient by a factor of e (2.5)

example from real atmosphere In 6 hours, temperature gradient doubles, a factor of 8 larger than that expected from scale analysis of geostrophic confluence Implication: ageostrophic non-QG forcing is important to the circulations on cross frontal scale QUASI-GEOSTROPHIC THEORY IS INSUFFICIENT TO ACCOUNT FOR THE VERTICAL MOTIONS IN THE VICINITY OF FRONTS