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heig

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Streamfunction

-1.4783

10 20 30 40

2

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12Buoyancy

-2.8188

H C WL H

H=High L=Low C=Cold W=Warm

gives vortex stretching and so intensifies the lower level cyclonic anomaly.

The streamfunction field tilts to the west with increasing height. From the omega

equation, there is ascent ahead of the cyclonic anomaly. This vertical motion

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JVAzh �kCEiR�G_O�Ti�FkABPN�T�kLjABD�LNF��BLj�^i�D����-�kCEi���ABFG�p�@DEFG�k]`�@PjiRaEi�M%]��XLNDT��n2]^aT� h�]z�^i^>pvRCTiRFG�G_Oi�]^���Â�@DEMY�kLjABD<Ei�PNa@Fw�kLNPj�kF��GA,�kCEiRi�]^FG�phVLj�kC�LNDEMY_ki�]^FOLNDT��CTi�Lj�BCt�%It]^DEa��kCTiR�Gi���~$i%_O]`�k�T_Oi <Ei�PNa��kLNPN�kFw�GA��kCTiRh�i�FG�%> � Lj�kC�kCTi"CEA^_OL#!%ABDt�k]^PRPNLNDTi3A^��A^�@FGi%_O��]`�kLjABD@F�]`���kCTidi�D@a ABDEPj�^I�h2i <@DEa��Â�G_OLj�XLN]^PNPN�T���kCE]`���kCEid_ki�Fk�EPN�kF�]`_kiiY�T]^MY�kPj� �kCEi3FO]^��id]^F �¸A^_��kCEi��^_OAzh�LNDE��n2]^aT�rh!]��^i^> q�DEFG�Gi�]^a A^��]�aEi�M%]��XLNDT��FGABPN�T�kLNABD�Ip]^F ~@]`_k�kPN�A^�@FGi%_O�^i�a�I@]��^_kA¼h�LNDT��FkABPN�T�kLjABD�LNF!A^�@�k]^LNDTi�a�IEaE�Ei��GA��kCTi !%i%_kA��@]^MkW��^_kAB�EDEa"FG�k]`�Gi^>vRCTi�iY�X~$i%_OLN��i�Dt�¥LNF�_ki%~$i�]`�Gi�a�IT�@�T�¥]�CTA^_OL"!%ABD��k]^P�PNL DTiVA^�#�@�TA¼�B]^DEMY�3LNF!A^�@FGi%_k�^i�a�]`��i%�^i%_k���kL ��i�Pji%�^i�P�kCTi�]^FOFkLN��LNP ]`�kLjABD3h�LNDEaEAzh,>pv�CTi,]^DE]^Pj�XFkLNFRL FRFkCTA¼h�DdLND #wLj�B�E_ki A�>�v�CTi & } $�]`_�FkMOCTi���i�LNF¥~E_OAz�XLNaTi�ah�Lj�kC�MYABD �@LNMY�kLNDT��LNDE�¸A^_O��]`�kLjABD�>³v�CTi�!%i%_kA��@]^MOWt�^_kAB�@DEa�FG�k]`�Gi���i�]^DEF��kCE]`���kCEiRFGABPN�T�kLjABD�FOCTAB�EPNa �^_kA¼h,ICTA¼h2i%�^i%_�Iw]�aTi�M%]z��LNDE�"FGABPN�T�kLNABD�LNF�A^�@FGi%_k�^i�a�>�vRCTi�]^DE]^Pj�XFkLNF�aTi�M%]z��F��¸A^_��kCTi <E_OFG��~5]`_k��A^�2�kCEi��kLN��ih�LND@aTAzh�]^DEad�^_kAzh�FR�¸A^_V�kCTi,FGi�MYABDEa"~@]`_k�%>vRCTi��^_kA¼hR�kC _O]`�Gi��kCT_kAB�E�BCTAB�T� �kCTi3�kLN��i�h�LNDEaTA¼h aEi%~[i�D@aEF�ABD �kCTi3_ki�P ]`�kLj�^i3h2i�Lj�BCt� �BLj�^i�D �GA��kCTi�@]^MOWt�^_kAB�@DEa FG�k]`�Gi^IR]^F�FkCTA¼h�D�LND #wLj�B�E_ki/x%{X> � Lj�kC�]rPN]`_k�^idh�i�Lj�BCt���BLj�^i�D �GA/�kCTidA^�@Fki%_k��]`�kLNABDEF%I�kCTi�]^D@]^Pj��FOLNF�]^PNFGAdaTi�M%]z��F��kCT_kAB�T�BCEAB�T���kCTi�]^DE]^PN��FkL F�h�LNDEaTA¼h,> � LN�kC/]dPN]`_k�^i�h�i�Lj�BCt�,�BLj�^i�Dr�GAd�kCTi�@]^MOWt�^_kAB�@DEa�FG�k]`�Gi^IX�kCTiV]^DE]^Pj�XFkLNF��^_kA¼h�F��kCE_kAB�T�BCTAB�T�p�kCTiRh�LND@aTAzh,> � Lj�kC�i�g��E]^PTh�i�Lj�BCt�kFph�iVFGi%i��kCE]`��kCTi�]^D@]^Pj��FOLNF!aTi�M%]��XF��kCE_kAB�T�BC��kCTi <@_OFG�¥~@]`_k�R]^D@a��^_kAzhVFR�kCT_kAB�E�BC3�kCTi�FGi�MYABDEad~@]`_k�¥A^�w�kCTi,h�L DEaTAzh�>

|`{

Page 22: NG %G) .&'! · hK> ^ > !kiN B=& > F " 9& lOF J&'>l G 8! b) & *! &' CBL H& > % =! Jo ' M OK& $ , I&'> 97aS hK> /GI " / ) ,& > /" IB

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Streamfunction

-0.97124

10 20 30 40-6

-4

-2

0

2

4

6

buoy

ancy

zonal direction

Bottom Buoyancy

10 20 30 40-6

-4

-2

0

2

4

6

buoy

ancy

zonal direction

Top Buoyancy

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Quasi-Geostrophic Potential Vorticity

0.0063373

σy-2 =1 σ

b-2 =2

LH

+N �/

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Streamfunction

-1.8363

10 20 30 40-6

-4

-2

0

2

4

6

buoy

ancy

zonal direction

Bottom Buoyancy

10 20 30 40-6

-4

-2

0

2

4

6

buoy

ancy

zonal direction

Top Buoyancy

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Quasi-Geostrophic Potential Vorticity

-0.00163080.0016308

σy-2=10 σ

b-2 =1

LH

+�� /

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Streamfunction

-1.5895

10 20 30 40-6

-4

-2

0

2

4

6

buoy

ancy

zonal direction

Bottom Buoyancy

10 20 30 40-6

-4

-2

0

2

4

6

buoy

ancy

zonal direction

Top Buoyancy

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Quasi-Geostrophic Potential Vorticity

-0.000228280.00022828

σy-2 =1000 σ

b-2 =1

L

+jÁ /

Analysis from 4D Var

Background StateTrue Solution (Observed on lower level)

#wLj�B�T_ki @8= +M��!�"�_9�����DBL��� � ! &`Oj� &�������"���i���"K! ����������"%!&������[OF��&�>n���x� ^ �����'ik! &������������[&'>��Y&'��?n"���i���"Q! ��7n!v������ ^ !�G*rNb��'������7,�`&*!&'��EuOF�J&'>n0 ! 5Md$tX�8GR&'��i ��"�_]�;�A��! "HOj����b >A&'��EH0 ^ 5 �:����b > &)b���i����,&��Y&�>���� ^ �����'i!&'��� ���M�%��B;@&����/���Db��'��!&'���)&�>=! �W&�>�� ^ !�G*rNb ��� ����7x�.&'! &��@OQ����b�>A&�EQ03G85��:����b >A&Db���i����x&'�,&�>��@� ^ �����'ik! &��������s�%� B;@?@?@<&'���/���b��'��! &�����&�>�!��/&�>�� ^ !�G*rNb��'�����=7D�.&'!&'�MOQ����b�>A&�S � &��'��!���BL����GR&'��� ��! ��7 % #�� # ;���"%7��/� � � "��%7�op?X� ���J&'��i �-GI� �A&������'��E7�! �.>��87UoK����bA!&���i��lGI���A&'�����*��S � ���_A! ��G�_<� � ��?�?X���s! �=7,"��OQ��� ^ � ����7�!��������/� # �������x��� &�&���7 o hC�'��� � � "��9&'��� �0�� ^ �.����i ��7�5IE � "����6�)!���>���7<o � !�G*rNb��'������7 � &*!&�� E��F�87 � ��"���7<of+M�=! "�_N���%�|Xx

Page 23: NG %G) .&'! · hK> ^ > !kiN B=& > F " 9& lOF J&'>l G 8! b) & *! &' CBL H& > % =! Jo ' M OK& $ , I&'> 97aS hK> /GI " / ) ,& > /" IB

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Streamfunction

-1.3615

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Quasi-Geostrophic Potential Vorticity

-0.2

4522

0.24522

0 10 20 30 40

2

4

6

8

10

12Buoyancy

-1.23221.6429

assimilation windowAnalysis at the beginning of the

Weights: Jb=1 Jo=1

C WL

+N  /

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Streamfunction

-2.7212

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Quasi-Geostrophic Potential Vorticity

-0.24570.2457

0 10 20 30 40

2

4

6

8

10

12Buoyancy

-4.7458

at this time - given by the black dots

Analysis at the end of the

Horizontal line of buoyancy is observed

L C WH H

assimilation window

+�� /#wLj�B�T_ki `,= # �'�OF����b:df!�7�_ �W!ki�� E$+M��!�"�_9�����DBL�'��� 2 � #$! �8S[+ >����'��v����A&'!�"f"������]� B-&�>�� ^ ���_A! ��G�_x;���"�7 ���� ^ �.����i ��7@!&Q&�>�������7���Ba&�>��s!��'�.���/��"%!&'��� ��OF����79�O6EAOF�J&'>]!Dv����'� ^ ! G*rNb��'������7@�.&'! &�� EN!���7@�%79���A&'�J&`_�G��ik!����%! �=GI��@!&'���%GI�8��S

|^|

Page 24: NG %G) .&'! · hK> ^ > !kiN B=& > F " 9& lOF J&'>l G 8! b) & *! &' CBL H& > % =! Jo ' M OK& $ , I&'> 97aS hK> /GI " / ) ,& > /" IB

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Streamfunction

-0.89504

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Quasi-Geostrophic Potential Vorticity

-0.19286

0.19286

0 10 20 30 40

2

4

6

8

10

12Buoyancy

-1.76591.7659

Analysis at the beginning of the

Weights: Jb=1 Jo=1

L H

W C

assimilation window

+N  /

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Streamfunction

-0.96296

0.48148

10 20 30 40

2

4

6

8

10

zonal direction

heig

ht

Quasi-Geostrophic Potential Vorticity

-0.19271

-0.12848

0 10 20 30 40

2

4

6

8

10

12Buoyancy

-1.9086

1.4315

Analysis at the end of theassimilation window

Observations of buoyancy at every timestep

L H

W C

+�� /#wLj�B�T_ki�A,= ����G�!k_A����b dj!�79_ �\!ki ��E�+M��!�"�_9������BL��� � 2 �Z#$!���S + >����'��v����A&*! "$"������Y� BF&'>�� ^ ���_ ! �=GI_ ;=��"%7[���� ^ �.����i ��7Y! &F��i�����_�&������6"���i ��"gE9OF�J&'>,!lv����'� ^ !�G*rNb��'������7��.&'!&'��E�!���7<��79���A&���&`_<GI�i!����%! ��G����@! &��'��G�����S

|�B

Page 25: NG %G) .&'! · hK> ^ > !kiN B=& > F " 9& lOF J&'>l G 8! b) & *! &' CBL H& > % =! Jo ' M OK& $ , I&'> 97aS hK> /GI " / ) ,& > /" IB

2 4 6 8 10 12 14 16 18 201

1.5

2

2.5

3

3.5

Time

Max

imum

am

plitu

de o

f the

buo

yanc

y fie

ld

Weights: Jb=1 Jo=10

+N �/

2 4 6 8 10 12 14 16 18 201

1.5

2

2.5

3

3.5

Time

Max

imum

am

plitu

de o

f the

buo

yanc

y fie

ld

Weights: Jb=1 Jo=1

+�� /

2 4 6 8 10 12 14 16 18 201

1.5

2

2.5

3

3.5

Time

Max

imum

am

plitu

de o

f the

buo

yanc

y fie

ld

Weights: Jb=10 Jo=1

+jÁ /

#wLj�B�T_ki3x%{;= # ���OK&'> �'! &��8�M� Bj2 � #$! �s! ��!�"�_9�����FBL� �-&'>��/79��G�!k_A����bYdj!�79_UOK!ki�� E=��������b<79��tu���'���A&�OQ����b�>A&�����bA�BL���K&'>��&0 � !���710 � &������@�K� B��/�A��! &������ H B S � ��"���7 �F�87�ojhC�'�9&�>[0L� ^ �����'i���7=5RE � "�����7�!���>���7Yof! ��!�"�_9����7aS

|�&

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